Understanding the Lomb-Scargle Periodogram

Jacob T. VanderPlas1

ABSTRACT

The Lomb-Scargle periodogram is a well-known algorithm for detecting and char-

acterizing periodic signals in unevenly-sampled data. This paper presents a conceptual

introduction to the Lomb-Scargle periodogram and important practical considerations

for its use. Rather than a rigorous mathematical treatment, the goal of this paper is

to build intuition about what assumptions are implicit in the use of the Lomb-Scargle

periodogram and related estimators of periodicity, so as to motivate important practical

considerations required in its proper application and interpretation.

Subject headings: methods: data analysis — methods: statistical

1. Introduction

The Lomb-Scargle periodogram (Lomb 1976; Scargle 1982) is a well-known algorithm for de-

tecting and characterizing periodicity in unevenly-sampled time-series, and has seen particularly

wide use within the astronomy community. As an example of a typical application of this method,

consider the data shown in Figure 1: this is an irregularly-sampled timeseries showing a single ob-

ject from the LINEAR survey (Sesar et al. 2011; Palaversa et al. 2013), with un-filtered magnitude

measured 280 times over the course of five and a half years. By eye, it is clear that the brightness

of the object varies in time with a range spanning approximately 0.8 magnitudes, but what is not

immediately clear is that this variation is periodic in time. The Lomb-Scargle periodogram is a

method that allows efficient computation of a Fourier-like power spectrum estimator from such

unevenly-sampled data, resulting in an intuitive means of determining the period of oscillation.

The Lomb-Scargle periodogram computed from this data is shown in the left panel of Figure 2.

The Lomb-Scargle periodogram here yields an estimate of the Fourier power as a function of period

of oscillation, from which we can read-off the period of oscillation of approximately 2.58 hours. The

right panel of Figure 2 shows a folded visualization of the same data as Figure 1 – i.e. plotted as

a function of phase rather than time.

Often this is exactly how the Lomb-Scargle periodogram is presented: as a clean, well-defined

procedure to generate a power spectrum and to detect the periodic component in an unevenly-

sampled dataset. In practice, however, there are a number of subtle issues that must be considered

1eScience Institute, University of Washington

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– 2 –

52750 53000 53250 53500 53750 54000 54250 54500time (MJD)

15.6

15.8

16.0

16.2

mag

nitu

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LINEAR object 11375941

Fig. 1.— Observed light curve from LINEAR object ID 11375941. Uncertainties are indicated by

the gray errorbars on each point. Python code to reproduce this figure, as well as all other figures

in this manuscript, is available at http://github.com/jakevdp/PracticalLombScargle/

when applying a Lomb-Scargle analysis to real-world datasets. Here are a few questions in particular

that we might wish to ask about the results in Figure 2:

1. How does the Lomb-Scargle periodogram relate to the classical Fourier power spectrum?

2. What is the source of the pattern of multiple peaks revealed by the Lomb-Scargle Peri-

odogram?

3. What is the largest frequency (i.e. Nyquist-like limit) that such an analysis is sensitive to?

4. How should we choose the spacing of the frequency grid for our periodogram?

5. What assumptions, if any, does the Lomb-Scargle approach make regarding the form of the

unknown signal?

6. How should we understand and report the uncertainty of the determined frequency?

Quantitative treatments of these sorts of questions are presented in various textbooks and review

papers, but I have not come across any single concise reference that gives a good intuition for how to

think about such questions. This paper seeks to fill that gap, and provide a practical, just-technical-

enough guide to the effective use of the Lomb-Scargle method for detection and characterization

of periodic signals. This paper does not seek a complete or rigorous treatment of the mathematics

involved, but rather seeks to develop the intuition of how to think about these questions, with

references to relevant technical treatments.

– 3 –

0.0 0.5 1.0 1.5 2.0 2.5Period (days)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Lom

b-S

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ower

Lomb-Scargle Periodogram

0.0 0.2 0.4 0.6 0.8 1.0phase

15.6

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16.0

16.2

mag

nitu

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Period = 2.58 hours

Phased Data

1 2 3 4 5Period (hours)

0.00.20.40.6

pow

er

Fig. 2.— Left panel: the Lomb-Scargle periodogram computed from the data shown in Figure 1,

with an inset detailing the region near the peak. Right panel: the input data in Figure 1, folded

over the detected 2.58-hour period to show the coherent periodic variability. For more discussion

of this particular object, see (Palaversa et al. 2013).

1.1. Why Lomb-Scargle?

Before we begin exploring the Lomb-Scargle periodogram in more depth, it is worth briefly

considering the broader context of methods for detecting and characterizing periodicity in time

series. First, it is important to note that there are many different modes of time series observation.

Point observation like that shown in Figure 1 is typical of optical astronomy: values (often with

uncertainties) are measured at discrete point in time, which may be equally or unequally-spaced.

Other modes of observation—e.g., time-tag-events, binned event data, time-to-spill events, etc.—

are common in high-energy astronomy and other areas. We will not consider such event-driven

data modes here, but note that there have been some interesting explorations of unified statistical

treatments of all of the above modes of observation (e.g. Scargle 1998, 2002).

Even limiting our focus to point observations, there are a large number of complementary

techniques for periodic analysis, which generally can be categorized into a few broad categories:

Fourier Methods are based on the Fourier transform, power spectra, and closely related correla-

tion functions. These methods include the classical or Schuster periodogram (Schuster 1898),

the Lomb-Scargle periodogram (Lomb 1976; Scargle 1982), the correlation-based method of

Edelson & Krolik (1988), and related approaches (see also Foster 1996, for a discussion of

wavelet transforms in this context).

Phase-folding Methods depend on folding observations as a function of phase, computing a

cost function across the phased data (often within bins constructed across phase space) and

optimizing this cost function across candidate frequencies. Some examples are String Length

– 4 –

(Dworetsky 1983), Analysis of Variance (Schwarzenberg-Czerny 1989), Phase Dispersion Min-

imization (Stellingwerf 1978), the Gregory-Laredo method (Gregory & Loredo 1992), and the

conditional entropy method (Graham et al. 2013a). Methods based on correntropy are similar

in spirit, but do not always require explicit phase folding (Huijse et al. 2011, 2012).

Least Squares Methods involve fitting a model to the data at each candidate frequency, and

selecting the frequency which maximizes the likelihood. The Lomb-Scargle periodogram also

falls in this category (see Section 5), as does the Supersmoother approach (Reimann 1994).

Other studies recommend statistics other than least square residuals; see, e.g., the orthogonal

polynomial fits of Schwarzenberg-Czerny (1996).

Bayesian Approaches apply Bayesian probability theory to the problem, often in a similar man-

ner to the phase-folding and/or least-squares approaches. Examples are the generalized Lomb-

Scargle models of Bretthorst (1988), the phase-binning model of Gregory & Loredo (1992),

Gaussian process models (e.g. Wang et al. 2012), and models based on stochastic processes

(e.g. Kelly et al. 2014).

Various reviews have been undertaken to compare the efficiency and effectiveness of the available

methods; for example, Schwarzenberg-Czerny (1999) focuses on the statistical properties of methods

and recommends those based on smooth model fits over methods based on phase binning, while

Graham et al. (2013b) instead take an empirical approach and find that when considering detection

efficiency in real datasets, no suitably efficient algorithm universally outperforms the others.

In light of this breadth of available methods, why limit our focus here to the Lomb-Scargle

periodogram? One reason is cultural: the Lomb-Scargle periodogram is perhaps the best-known

technique to compute periodicity of unequally-spaced data in astronomy and other fields, and so

is the first tool many will reach for when searching for periodic content in a signal. But there is a

deeper reason as well; it turns out that Lomb-Scargle occupies a unique niche: it is motivated by

Fourier analysis (see Section 5), but it can also be viewed as a least squares method (see Section 6).

It can be derived from the principles of Bayesian probability theory (see Section 6.5), and has been

shown to be closely related to bin-based phase-folding techniques under some circumstances (see

Swingler 1989). Thus, the Lomb-Scargle periodogram occupies a unique point of correspondence

between many classes of methods, and so provides a focus for discussion that has bearing on

considerations involved in all of these methods.

1.2. Outline

The remainder of this paper is organized as follows:

Section 2 presents a review of the continous Fourier transform and some of its useful properties,

including defining the notion of a power spectrum (i.e. classical/Schuster periodogram) for detect-

ing periodic content in a signal.

– 5 –

Section 3 builds on these properties to explore how different observation patterns (i.e. window

functions) affect the periodogram, and discusses a conceptual understanding of the Nyquist limiting

frequency.

Section 4 considers non-uniformly sampled signals as a special case of an observational window,

and shows that the Nyquist-like limit for this case is quite different than what many practitioners

in the field often assume.

Section 5 introduces the Lomb-Scargle periodogram, a modified version of the classical peri-

odogram for unevenly-sampled data, as well as the motivation behind these modifications.

Section 6 discusses a complementary view of the Lomb-Scargle periodogram as the result of least-

squares model fitting, as well as some extensions enabled by that viewpoint.

Section 7 builds on the concepts introduced in earlier sections to discuss important practical con-

siderations for the use of the Lomb-Scargle periodogram, including the choice of frequency grid,

uncertainties and false alarm probabilities, and modes of failure that should be accounted for when

working with real-world data.

Section 8 concludes and summarizes the key recommendations for practical use of the Lomb-

Scargle periodogram.

2. Background: the Continuous Fourier Transform

In order to understand how we should interpret the Lomb-Scargle periodogram, we will first

briefly step back and review the subject of Fourier analysis of continuous signals. Consider a

continuous signal g(t). Its Fourier transform is given by the following integral, where i ≡√−1

denotes the imaginary unit:

g(f) ≡∫ ∞−∞

g(t)e−2πiftdt (1)

The inverse relationship is given by:

g(t) ≡∫ ∞−∞

g(f)e+2πiftdf (2)

For convenience we will also define the Fourier transform operator F , such that

F{g} = g (3)

F−1{g} = g (4)

The functions g and g are known as a Fourier pair, which we will sometimes denote as g ⇐⇒ g.

2.1. Properties of the Fourier Transform

The continuous Fourier transform has a number of useful properties that we will make use of

in our discussion.

– 6 –

The Fourier transform is a linear operation. That is, for any constant A and any functions

f(t) and g(t), we can write:

F{f(t) + g(t)} = F{f(t)}+ F{g(t)}F{Af(t)} = AF{f(t)} (5)

Both identities follow from the linearity of the Fourier integral.

The Fourier transform of sinusoid with frequency f0 is a sum of delta functions at ±f0.

From the integral definition of the Dirac delta function1, we can write

F{e2πf0t} = δ(f − f0). (6)

Using Euler’s formula2 for the complex exponential along with the linearity of the Fourier

transform leads to the following identities:

F{cos(2πf0t)} =1

2[δ(f − f0) + δ(f + f0)]

F{sin(2πf0t)} =1

2i[δ(f − f0)− δ(f + f0)] . (7)

In other words, a sinusoidal signal with frequency f0 has a Fourier transform consisting of a

weighted sum of delta functions at ±f0.

A time-shift imparts a phase in the Fourier transform. Given a well-behaved function g(t)

we can use a transformation of variables to derive the following identity:

F{g(t− t0)} = F{g(t)}e−2πift0 (8)

Notice that the time-shift does not change the amplitude of the resulting transform, but only

the phase.

These properties taken together make the Fourier transform quite useful for the study of periodic

signals. The linearity of the transform means that any signal made up of a sum of sinusoidal

components will have a Fourier transform consisting of a sum of delta functions marking the

frequencies of those sinusoids: that is, the Fourier transform directly measures periodic content

in a continuous function.

Further, if we compute the squared amplitude of the resulting transform, we can both do away

with complex components and remove the phase imparted by the choice of temporal baseline; this

squared amplitude is usually known as the power spectral density or simply the power spectrum:

Pg ≡ |F{g}|2 (9)

1 δ(f) ≡∫∞−∞ e

−2πixfdf

2eix = cosx+ i sinx

– 7 –

The power spectrum of a function is a positive real-valued function of the frequency f that quantifies

the contribution of each frequency f to the total signal. Note that if g is real-valued, it follows that

Pg is an even function; i.e. Pg(f) = Pg(−f).

2.2. Some Useful Fourier Pairs

(a) 1/f0

Sinusoid

+f0f0

Delta Functions

(b)

Gaussian

(2 ) 1

Gaussian

(c) T

Top Hat

2/T

Sinc

(d) T

Dirac Comb

1/T

Dirac Comb

Fig. 3.— Visualization of important Fourier pairs.

We have already discussed that the Fourier transform of a complex exponential is a single delta

function. This is just one of many Fourier pairs to keep in mind as we progress to understanding

the Lomb-Scargle periodogram. We list a few more important pairs here (see Figure 3 for a visual

representation of the following pairs):

The Fourier transform of a sinusoid is a pair of Delta functions. (See Figure 3a)

F{cos(2πf0t)} =1

2[δ(f − f0) + δ(f + f0)] (10)

We saw this above, but repeat it here for completeness.

The Fourier transform of a Gaussian is a Gaussian. (See Figure 3b)

F{N(t;σ)} =1√

2πσ2N

(f ;

1

2πσ

)(11)

– 8 –

The Gaussian function N(t, σ) is given by

N(t;σ) ≡ 1√2πσ2

e−t2/(2σ2) (12)

The Fourier transform of a rectangular function is a sinc function. (See Figure 3c)

F{ΠT (t)} = sinc(fT ) (13)

The rectangular function, Π(t), is a normalized symmetric function that is uniform within a

range given by T , and zero elsewhere:

ΠT (t) ≡

{1/T, |t| ≤ T/20, |t| > T/2

(14)

The sinc function is given by the standard definition:

sinc(x) ≡ sin(πx)

πx(15)

The Fourier transform of a Dirac comb is a Dirac comb. (See Figure 3d)

F{IIIT (t)} =1

TIII1/T (f) (16)

The Dirac comb, IIIT (t), is an infinite sequence of Dirac delta functions placed at even

intervals of size T :

IIIT (t) ≡∞∑

n=−∞δ(t− nT ) (17)

Notice in each of these Fourier pairs the reciprocity of scales between a function and its Fourier

transform: a narrow function will have a broad transform, and vice versa. More quantitatively, a

function with a characteristic scale T will in general have a Fourier transform with characteristic

scale of 1/T . Such reciprocity is central to many diverse applications of Fourier analysis, from

electrodynamics to music theory to quantum mechanics. For our purposes, this property will turn

out to be quite important as we push further in understanding the Lomb-Scargle periodogram.

2.3. The Convolution Theorem

A final property of the Fourier transform that we will discuss here is its ability to convert

convolutions into point-wise products. A convolution of two functions, usually denoted by the ∗symbol, is defined as follows:

[f ∗ g](t) ≡∫ ∞−∞

f(τ)g(t− τ)dτ (18)

– 9 –

0

2

4

6

8

10

0.0 0.2 0.4 0.6 0.8 1.00

2

4

6

8

10

Fig. 4.— Visualization of a convolution between a continuous signal and a rectangular smoothing

kernel. The normalized rectangular window function slides across the domain (upper panel), such

that at each point the mean of the values within the window are used to compute the smoothed

function (lower panel).

From the definition, it is clear that a convolution amounts to “sliding” one function past the other,

integrating at each step. Such an operation is commonly used, for example, in smoothing a function,

as visualized in Figure 4 for a rectangular smoothing window.

Given this definition of a convolution, it can be shown that the Fourier transform of a convo-

lution is the point-wise product of the individual Fourier transforms:

F{f ∗ g} = F{f} · F{g} (19)

In practice, this can be a much more efficient means of computing a convolution than to directly

solve at each time t the integral over τ that appears in Equation 18. The identity in Equation 19 is

known as the convolution theorem, and is illustrated in Figure 5. An important corrollary is that

the Fourier transform of a product is a convolution of the two transforms:

F{f · g} = F{f} ∗ F{g} (20)

We will see that these properties of the Fourier transform become essential when thinking about

frequency components of time-domain measurements.

– 10 –

5

0

5 Data: D(t) Data FT: (D)

0

5

10Window: W(t) Window FT: (W)

0.2 0.4 0.6 0.8t

4

2

0

2

4[D W](t) = 1{ [D] [W]}

30 20 10 0 10 20 30f

(D) (W)

Convolution Pointwise Product

FT

FT

FT

Fig. 5.— Visualization of the convolution theorem (Equation 19). Recall that the Fourier transform

of a convolution is the pointwise product of the two Fourier transforms. In the right panels, the

black and gray lines represent the real and imaginary part of the transform, respectively.

3. Window Functions: From Idealized to Real-world signals

Until now we have been discussing Fourier transforms of continuous signals which are well-

defined for all times −∞ < t < ∞. Real-world temporal measurements of a signal, however, only

involve some finite span of time, at some finite rate of sampling. In either case, the resulting

data can be described by a point-wise product of the true underlying continuous signal with a

window function describing the observation. For example, a continuous signal measured over a finite

duration is described by a rectangular window function spanning the duration of the observation,

and a signal measured at regular intervals is described by a Dirac comb window function marking

those measurement times.

The Fourier transform of measured data in these cases, then, is not the transform of the

continuous underlying function, but rather the transform of the point-wise product of the signal

and the observing window. Symbolically, if the signal is g(t) and the window is W (t), the observed

function is

gobs(t) = g(t)W (t), (21)

and by the convolution theorem, its transform is a convolution of the signal transform and the

– 11 –

window transform:

F{gobs} = F{g} ∗ F{W}. (22)

This has some interesting consequences for the use and interpretation of periodograms, as we shell

see.

3.1. Effect of a Rectangular Window

2

1

0

1

2

Signal

0.0

0.3

0.6

0.9

1.2

Window

6 4 2 0 2 4 6time

2

1

0

1

2

4 2 0 2 4frequency

Observed

ConvolutionPointwise Product

FT

FT

FT

Fig. 6.— Visualization of the effect on the Fourier transform of a rectangular observing window

(i.e., a continuous signal observed in its entirety within a finite range of time). The function used

here is g(t) = 1.2 sin(2πt)+0.8 sin(4πt)+0.4 sin(6πt)+0.1 sin(8πt); The observed Fourier transform

is a convolution of the true transform (here a series of Delta functions indicating the component

frequencies) and the window transform (here a narrow sinc function).

First, let’s consider the case of observing a continuous periodic signal over a limited span of

time: Figure 6 shows a continuous periodic function observed only within the window −3 < t < 3.

The observed signal in this case can be understood as the pointwise product of the underlying

infinite periodic signal with a rectangular window function. By the convolution theorem, the Fourier

transform will be given by the convolution of the transform of the underlying function (here a set

of delta functions at the component frequencies) and the transform of the window function (here

a sinc function). For the purely periodic signal like the one seen in Figure 6, this convolution has

– 12 –

the effect of replacing each delta function with a sinc function. Because of the inverse relationship

between the width of the window and the width of its transform (see Figure 3), it follows that a

wider observing window leads to proportionally less spread in the Fourier transform of the observed

function.

3.2. The Dirac Comb and the Discrete Fourier Transform

0.0

0.5

1.0

Signal

0.0

0.5

1.0 Window

4 2 0 2 4time

0.0

0.5

1.0

t = 0.5

4 2 0 2 4frequency

Observed

ConvolutionPointwise Product

FT

FT

FT

Fig. 7.— Visualization of the effect on the Fourier transform of a Dirac Comb observing window

(i.e., a long string of evenly-spaced discrete observations). The observed Fourier transform is a

convolution of the true transform (here a localized Gaussian) and the window transform (here

another Dirac comb).

Another window function that commonly arises is when a continuous signal is sampled (nearly)

instantaneously at regular intervals. Such an observation can be thought of as a point-wise product

between the true underlying signal and a Dirac comb with the T parameter matching the spacing

of the observations; this is illustrated in Figure 7. Interestingly, because the Fourier transform of

a Dirac comb is another Dirac comb, the effect of such an observing window is to create a long

sequence of aliases of the underlying transform with a spacing of 1/T . With this in mind, we can be

assured in this case that evaluating the observed transform in the range 0 ≤ f < 1/T is sufficient

to capture all the available frequency information: the signal outside that range is a sequence of

– 13 –

identical aliases of what lies within that range.

3.2.1. The Nyquist Limit

The example in Figure 7 is somewhat of a best-case scenario, because the true Fourier transform

values are non-zero only within a range of width 1/T . If we increase the time between observations,

decreasing the spacing of the frequency comb, the true transform no longer “fits” inside the window

transform, and we will have a situation similar to that in Figure 8. The result is a mixing of different

portions of the signal, such that the true Fourier transform cannot be recovered from the transform

of the observed data!

0.0

0.5

1.0

Signal

0.0

0.5

1.0 Window

4 2 0 2 4time

0.0

0.5

1.0

t = 1.5

4 2 0 2 4frequency

Observed

ConvolutionPointwise Product

FT

FT

FT

Fig. 8.— Repeating the visualization from Figure 7, but here with a lower sampling rate. The result

is that the Fourier transform of the window function (middle right) has spacing narrower than the

Fourier transform of the signal (upper right), meaning the observed Fourier transform (lower right)

has aliasing of signals, such that not all frequency information can be recovered. This is the reason

for the famous Nyquist sampling theorem, which conceptually says that only a function whose

Fourier transform can fit entirely between the “teeth” of the comb is able to be fully recovered for

regularly-spaced observations.

This implies that if we have a regularly-sampled function with a sampling rate of f0 = 1/T ,

we can only fully recover the frequency information if the signal is band-limited between frequencies

– 14 –

±f0/2. This is one way to motivate the famous Nyquist sampling limit, which approaches the

question from the other direction and states that to fully represent the frequency content of a

“band-limited signal” whose Fourier transform is zero outside the range ±B, we must sample the

data with a rate of at least fNy = 2B.

3.2.2. The Discrete Fourier Transform

When a continuous function is sampled at regular intervals, the delta functions in the Dirac

comb window serve to collapse the Fourier integral into a Fourier sum, and in this manner we can

arrive at the common form of the discrete Fourier transform. Suppose we have a true (infinitely

long and continuous) signal g(t), but we observe it only at a regular grid with spacing ∆t. In this

case, our observed signal is gobs = g(t) III∆t(t) and its Fourier transform is

gobs(f) =

∞∑n=−∞

g(n∆t)e−2πifn∆t, (23)

which follows directly from Equation 1 and Equation 17.

In the real world, however, we will not have an infinite number of observations, but rather

a finite number of samples N . We can choose the coordinate system appropriately and define

gn ≡ g(n∆t) to write

gobs(f) =N∑n=0

gne−2πifn∆t (24)

From the arguments around Nyquist aliasing, we know that the only relevant frequency range is

from 0 ≤ f ≤ 1/∆t, and so we can define N evenly-spaced frequencies with ∆f = 1/(N∆t) covering

this range. Denoting the sampled transform as gk ≡ gobs(k∆f), we can write

gk =N∑n=0

gne−2πikn/N (25)

which you might recognize as the standard form of the discrete Fourier transform.

Notice, though, that we glossed over one important thing: the effect of switching from an infi-

nite number of samples to a finite number of samples. In moving from Equation 23 to Equation 24,

we have effectively applied to our data a rectangular window function of width N∆t. From the

discussion accompanying Figure 6, we know what this does: it gives us a Fourier transform con-

volved with a sinc function of width 1/(N∆t), resulting in the “smearing” of the Fourier transform

signal with this width. Roughly speaking, then, any two Fourier transform values at frequencies

within 1/(N∆t) of each other will not be independent, and so we should space our evaluations of

the frequency with ∆f ≥ 1/(N∆t). Comparing to above, we see that this is exactly the frequency

spacing we arrived at from Nyquist-frequency arguments.

– 15 –

What this indicates is that the frequency spacing of the discrete Fourier transform is optimal

in terms of both the Nyquist sampling limit and the effect of the finite observing window! Now,

this argument has admittedly been a bit hand-wavy, but there do exist mathematically rigorous

approaches to proving that the discrete Fourier transform in Equation 25 captures all of the available

frequency information for a uniformly-sampled function gn (see, e.g. Vetterli et al. 2014). Despite

our lack of rigor here, I find this to be a helpful approach in developing intuition regarding the

relationship between the continuous and discrete Fourier transforms.

3.3. The Classical Periodogram

With the discrete Fourier transform defined in Equations 24-25, we can apply the definition

of the Fourier power spectrum from Equation 9 to compute the classical periodogram, sometimes

called the Schuster periodogram after Schuster (1898) who first proposed it:

PS(f) =1

N

∣∣∣∣∣N∑n=1

gne−2πiftn

∣∣∣∣∣2

(26)

Apart from the 1/N proportionality, this sum is precisely the Fourier power spectrum in Equation 9,

computed for a continuous signal observed with uniform sampling defined by a Dirac comb. It

follows that, in the uniform sampling case, the Schuster periodogram captures all of the relevant

frequency information present in the data. This definition readily generalizes to the non-uniform

case, which we will explore in the following section.

One point that should be emphasized is that the periodogram in Equation 26 and the power

spectrum in Equation 9 are conceptually different things. As noted in Scargle (1982), the astronomy

community tends to use these terms interchangeably, but to be precise the periodogram—i.e., the

statistic we compute from our data—is an estimator of the power spectrum—i.e., the underlying

continuous function of interest. In fact, the classical periodogram and its extensions (including the

Lomb-Scargle we will discuss momentarily) are not consistent estimators of the power spectrum:

that is, the periodogram has unavoidable intrinsic variance, even in the limit of an infinite number

of observations (for a detailed discussion, see Chp 8.4 of Anderson 1971).

4. Non-uniform Sampling

In the real world, particularly in fields like Astronomy where observations are subject to

influences of weather and diurnal, lunar, or seasonal cycles, the sampling rate is generally far from

uniform. Using the same approach as we used to explore uniform sampling in the previous section,

we can now explore non-uniform sampling here.

In the general non-uniform case, we measure some signal at a set of N times which we will

– 16 –

denote {tn}, which leads to the following observing window:

W{tn}(t) =N∑n=1

δ(t− tn) (27)

Applying this window to our true underlying signal g(t), we find an observed signal of the form:

gobs(t) = g(t)W{tn}(t)

=

N∑n=1

g(tn)δ(t− tn) (28)

Just as in the evenly-sampled case, the Fourier transform of the observed signal is a convolution

of the transforms of the true signal and the window:

F{gobs} = F{g} ∗ F{W{tn}} (29)

Unlike in the uniform case, the window transform F{W{tn}} will generally not be a straightforward

sequence of delta functions; the symmetry present in the Dirac comb is broken by the uneven

sampling, leading the transform to be much more “noisy”. This can be seen in Figure 9, which

shows the Fourier transform of a non-uniform observing window with an average sampling rate

identical to that in Figure 7, along with its imact on the observed Fourier transform.

A few things stand-out in this figure. In particular, the Fourier transform of the non-uniformly

spaced delta functions looks like random noise, and in some sense it is: the locations and heights of

the Fourier peaks are related to the intervals between observations, and so randomization of obser-

vation times leads to a randomization of Fourier peak locations and heights. That is, non-structured

spacing of observations will lead to a non-structured frequency peaks in the window transform. This

non-structured window transform, when convolved with the Fourier transform of the true signal,

results in an observed Fourier transform reflecting the same random noise. Comparing to the

uniformly-spaced observations in Figure 7, we see that the unstructured nature of the window

transform means that there is no exact aliasing of the true signal, and thus no way to exactly

recover any portion of the true Fourier transform for the underlying function.

One might hope that sampling the signal more densely might alleviate these problems, and

it does, but only to a degree. In Figure 10 we increase the density of observations by a factor of

10, such that there are 200 total observations over the length-10 observing window. The observed

Fourier transform in this case is much more reflective of the underlying signal, but still contains

a degree of “noise” rooted in the randomized frequency peaks due to randomized spacing between

observations.

4.1. A Non-uniform Nyquist Limit?

We saw in Section 3.2.1 that the Nyquist limit is a direct consequence of the symmetry in the

Dirac comb window function that describes evenly-sampled data, and uneven sampling destroys

– 17 –

0.0

0.5

1.0

Signal

0.0

0.5

1.0 Window

4 2 0 2 4time

0.0

0.5

1.0

avg( t) = 0.5

4 2 0 2 4frequency

Observed

ConvolutionPointwise Product

FT

FT

FT

Fig. 9.— The effect of non-uniform sampling on the observed Fourier transform. These samples

have the same average spacing as those in Figure 7, but the irregular spacing within the observing

window translates to irregular frequency peaks in its transform, causing the observed transform

to be “noisy”. Here black and gray lines represent the real and imaginary parts of the transform,

respectively.

the symmetry that underlies its definition. Nevertheless, the idea of the “Nyquist frequency” seems

to have taken hold in the scientific psyche to the extent that the idea is often mis-applied in areas

where it is mathematically irrelevant. For unevenly-sampled data, the truth is that the “Nyquist

limit” might or might not exist, and even in cases where it does exist it tends to be far larger (and

thus far less relevant) than in the evenly-sampled case.

4.1.1. Incorrect Limits in the Literature

In the scientific literature it is quite common to come across various proposals for a Nyquist-like

limit applied in the case of irregular sampling. A few typical approaches include using the mean of

the sampling intervals (e.g. Scargle 1982; Horne & Baliunas 1986; Press et al. 2007), the harmonic

mean of the sampling intervals (e.g. Debosscher et al. 2007), the median of the sampling intervals

(e.g. Graham et al. 2013b), or the minimum sample spacing (e.g. Percy 1986; Roberts et al. 1987;

Press & Rybicki 1989; Hilditch 2001). All of these “pseudo-Nyquist” limits are tempting criteria in

– 18 –

0.0

0.5

1.0

Signal

0.0

0.5

1.0 Window

4 2 0 2 4time

0.0

0.5

1.0

avg( t) = 0.05

4 2 0 2 4frequency

Observed

ConvolutionPointwise Product

FT

FT

FT

Fig. 10.— The effect of non-uniform sampling on the observed Fourier transform, with a factor of 10

more samples than Figure 9. Even with very dense sampling of the function, the Fourier transform

cannot be exactly recovered due to the imperfect aliasing present in the window transform.

that they are easy to compute, and reduce to the classical Nyquist frequency in the limit of evenly-

spaced data. Unfortunately, none of these approaches is correct: in general, unevenly-sampled data

can probe frequencies far larger than any of these supposed limits (a fact that several of these

citations do hint at parenthetically).

As a simple example of where such logic can fail spectacularly, consider the data from Fig-

ure 1: though the mean sample spacing is one observation every seven days, in Figure 2 we were

nevertheless able to quite clearly identify a period of 2.58 hours— an order of magnitude shorter

than the average-based pseudo-Nyquist limit would indicate as possible. For the data in Figure 1,

the minimum sample spacing is just under 10 seconds, but it would be irresponsible to claim that

this single pair of observations by itself defines some limit beyond which frequency information is

unattainable.

As a more extreme example, consider the data shown in Figure 11. This consists of noisy

samples from a sinusoid with a period of 0.01 units, with sample spacings ranging between 2 and

18 units: needless to say, any pseudo-Nyquist definition based on an average or minimum sample

spacings will be far below the true frequency of 100; still, the the Lomb-Scargle periodogram in the

– 19 –

200 400 600 800 10002.5

5.0

7.5

10.0

12.5

15.0

17.5Input Data

0.0 0.2 0.4 0.6 0.8 1.02.5

5.0

7.5

10.0

12.5

15.0

17.5Phased Data (Period=0.01)

2.5 5.0 7.5 10.0 12.5 15.0 17.5t

0

2

4

6

N(

t)

Intervals between Observations

10 2 10 1 100 101 102

frequency

0.0

0.2

0.4

0.6

0.8

1.0

Lom

b-S

carg

le P

ower

0.5/

mea

n(t)

mea

n(0.

5/t)

0.5/

min

(t)

Periodogram

Fig. 11.— An example of data for which the various poorly-motivated “pseudo-Nyquist” approaches

outlined in Section 4.1 fail spectacularly. The upper panels show the data, a noisy sinusoid with a

frequency of 100 (i.e. a period of 0.01). The lower left panel shows a histogram of spacings between

observations: the minimum spacing is 2.55, meaning that the signal has over 250 full cycles between

the closest pair of observations. Nevertheless, the periodogram (lower right) clearly identifies the

correct period, though it is orders of magnitude larger than pseudo-Nyquist estimates based on

average or minimum sampling rate.

lower right panel quite cleanly recovers the true frequency.

4.1.2. The Non-uniform Nyquist Limit

While pseudo-Nyquist arguments based on average or minimum sampling fail spectacularly,

there is a sense in which the Nyquist limit can be applied to unevenly-spaced data. Eyer & Bartholdi

(1999) explore this issue in some detail, and in particular prove the following:

Let p be the largest value such that each ti can be written ti = t0 + nip, for integers ni.

The Nyquist Frequency then is fNy = 1/(2p).

In other words, computing the Nyquist limit for unevenly-spaced data requires finding the largest

– 20 –

factor p, such that each spacing ∆ti is exactly an integer multiple of this factor. Eyer & Bartholdi

(1999) prove this formally, but the result can be understood by thinking of such data as a windowed

version of uniformly sampled data with spacing p, where the window is zero at all points but the

location of the observations. Such uniform data has a classical Nyquist limit of 1/(2p), and a

window function applied on top of that sampling does not change that fact.

Figure 12 shows an example of such a Nyquist frequency. The data are non-uniformly sampled

at times ti = ni · p, with p = 0.01 and ni drawn randomly from positive integers less than 10,000.

According to the Eyer & Bartholdi (1999) definition, this results in a Nyquist frequency fNy = 50,

and we see the expected behavior beyond this frequency: the signal at f > fNy consists of a series

of exact aliases of the signal at |f | < fNy.

0 20 40 60 80 100t

4

6

8

10

12

14

16

Non-uniform sampling; t = p * 0.01

0 50 100 150 200 250 300frequency

0.0

0.2

0.4

0.6

0.8

1.0Lomb-Scargle periodogram

Fig. 12.— A visualization of the Eyer & Bartholdi (1999) definition of the Nyquist frequency. Data

are non-uniformly sampled at times ti = nip, for integer ni and p = 0.01. This results in a Nyquist

frequency of fNy = (2p)−1 = 50: the periodogram outside the range 0 ≤ f < fNy is a series of

perfect aliases of the signal within that range.

We should keep in mind one consequence of this Nyquist definition: if you have any pair of

observation spacings whose ratio is irrational, the Nyquist limit does not exist!. To realize this

situation in practice, however, would require infinitely precise measurements of the times ti; finite

precision of time measurements means the Nyquist frequency can be large, but not infinite. For

example, if your observation times are recorded to D decimal places, the Nyquist frequency will be

at most

fNy ≤1

210D, (30)

with the inequality due to the fact that larger common factors may exist. In other words, absent

other relevant patterns in the observations, the Nyquist frequency for irregularly-sampled data is

most typically set by the precision of the time measurements (see also Bretthorst 2003; Koen 2006,

for more rigorous treatments of this result).

– 21 –

4.1.3. Frequency Limit due to Windowing

In contexts where observations are not instantaneous, but rather consist of short-duration

integrations of a continuous signal, a qualitatively different kind of frequency limit exists. This is

typical in, e.g., optical astronomy, where a single observation typically consists of an integration

of observed photons over a finite duration δt. As noted by Ivezic et al. (2014), this time-scale

of integration represents another kind of limiting frequency for irregularly-sampled data. Such a

situation means that the observation is effectively a convolution of the underlying signal with a

rectangular window function of width δt, in a manner analogous to Figure 5. By the convolution

theorem, the observed transform will be a point-wise product between the true transform and the

transform of the window, which will generally have a width proportional to 1/δt. This means that—

absent other more constraining window effects—the frequency limit is fmax ∝ 1/(2δt), with the

constant of proportionality dependent on the shape of the effective window describing individual

observations.

Keep in mind that the windowing limit 1/(2δt) is quite different than a Nyquist limit: the

Nyquist limit is the frequency beyond which all signal is aliased into the Nyquist range; the win-

dowing limit is the frequency beyond which all signal is attenuated to zero. In practice, the limit

implied by either the temporal resolution or windowing of individual observations may be too large

to be computationally feasible; for discussion of frequency limits in practice, see Section 7.1.

4.2. Semi-structured Observing Windows

We have seen that for uniform data, the perfect aliasing beyond the Nyquist frequency is a

direct consequence of the symmetry of the Dirac-comb window function. For non-uniform obser-

vations, such symmetry does not exist, but structure in the observing window can lead to partial

aliasing of signals in the data (see, e.g. Deeming 1975). In this section, we will examine two typi-

cal window functions derived from real-world observations: one ground-based (LINEAR) and one

space-based (Kepler). For details on how window power spectra can be estimated in practice, see

Section 7.3.

4.2.1. A Ground-based Observing Window: LINEAR

Let’s again consider the data shown in Figure 1. The window power spectrum for this in

Figure 13 shows some quite distinct features, and these features have an intuitive interpretation.

Namely, if the window power shows a spike at a period of p days, this means that an observation

at time t0 is likely to be followed by another observation near a time t0 + np for integer n.

With this in mind, the strong spike at a period of 1 day indicates that observations are taken

near the same time of day: this is typical of a ground-based survey with observations recorded only

– 22 –

10 1 100 101 102 103

period

0.0

0.2

0.4

0.6

0.8

1.0

win

dow

pow

er

Fig. 13.— The power spectrum of the observing window for the data shown in Figure 1. Notice

the strong spike in power at a period of 1 day, and related aliases at 1/n days for integer n. There

is also a strong spike at 365 days, and a noticeable spike at ∼ 14 days. Each of these indicate time

intervals that appear often in the data.

during the nighttime hours. The additional spikes at periods of 1/n days (for integer n) are aliases

of this same feature. Figure 13 also shows a wide spike at a period of 1 year, indicating a detectable

annual pattern in the observations. Finally, there is a noticeable spike at approximately 14 days

that is likely related to patterns of scheduling within the survey.

Recall that a Fourier spectrum observed through a particular window will reflect a convolution

of the true spectrum and the window spectrum (cf. Figure 9), and so we would expect the structure

in the window to be imprinted on the power spectrum measured from the data.

This imprint of the window power is illustrated in Figure 14. The upper panel shows the

window power spectrum as a function of frequency (rather than period, as in Figure 13), while

lower panel shows the observed signal power spectrum as a function of frequency (rather than

period, as in Figure 2). The upper panel and its inset show clearly the 1-day and 1-year features we

noted previously. The lower panel shows the observed power spectrum of the data: these diurnal and

annual peaks in the window function are quite clearly imprinted on the observed power spectrum at

relevant scales. This approximate aliasing is similar to the exact aliasing seen in regularly-sampled

data at the Nyquist frequency; however, in this case the magnitude of the aliased signal fades

further from the frequency driving the signal.

4.2.2. A Space-based Observing Window: Kepler

Space-based surveys will generally have quite different observing windows. For example, the

upper-left panel of Figure 15 shows observations of an RR-Lyrae variable star from the Kepler

– 23 –

0.0

0.2

0.4

0.6

0.8

1.0

Lom

b-S

carg

le p

ower

Window Power Spectrum

0 5 10 15 20 25frequency (days 1)

0.0

0.2

0.4

0.6

0.8

1.0

Lom

b-S

carg

le p

ower

Light Curve Power Spectrum

0 1 2 3 4 5frequency (years 1)

0.00

0.25

0.50

0.75

1.00

2 1 0 1 2frequency - fmax (years 1)

0.00

0.25

0.50

0.75

1.00

Fig. 14.— The effect of the window function in Figure 13 on the power spectrum in Figure 2.

The top panel shows the window power spectrum, and the bottom panel shows the observed signal

power spectrum. Both are plotted as a function of frequency (we earlier saw both of these as

a function of period; see Figure 13 and Figure 2, respectively). Viewing these as a function of

frequency makes it clear that the structure in the window function is imprinted on the observed

spectrum: both the diurnal structure in the main panel, and the annual structure in the inset are

apparent in the observed spectrum.

survey, measured 4083 times over a period of three months, with an irregular observing cadence of

around 30 minutes (For deeper discussion of these observations, see Kolenberg et al. 2010). The

Kepler observations are very nearly uniformly-spaced, and this is reflected in the window power

spectrum, shown in the upper-right panel of Figure 15. The window function is a series of very

narrow evenly-spaced spikes, reminiscent of the Dirac comb shown in Figures 7-8. By analogy we

can treat fNy = 0.5/29.4 minutes−1 as the effective Nyquist limit for the data, keeping in mind

that aliasing beyond this “limit” will be imperfect due to the uneven spacing of the samples (see

the lower-left panels of Figure 15). The lower-right panel of Figure 15 shows the power spectrum

of the observations, with gray shading indicating the (nearly) aliased region of the spectrum. The

period of 13.6 hours is quite strongly apparent, along with smaller spikes at integer multiples of

this frequency that indicate higher-order periodic components in the signal.

The window functions for ground-based and space-based observations, reflected by LINEAR

– 24 –

180 200 220 240 260

10

12

14

16

SA

P fl

ux (1

06 e/s

)

Observed light curve

0 1 2 3 4 5 6

0.00

0.25

0.50

0.75

1.00

Lom

b-S

carg

le p

ower

(29.4 minutes) 1

Window Power Spectrum

101

103

t (m

in)

Time between observations

0.0 0.5 1.0 1.5 2.0frequency (hours 1)

0.00

0.25

0.50

0.75

1.00

Lom

b-S

carg

le p

ower (Aliased Region)fmax = (13.60 hours) 1

Light Curve Power Spectrum

180 200 220 240 260Observation time (days)

800

80

tp W

(ms)

Kepler object ID 007198959

Fig. 15.— An RR Lyrae variable observed by the Kepler project (see Kolenberg et al. 2010). upper

left: the 4083 observed fluxes over roughly three months. upper right: the window power spectrum,

which is quite close to that of regularly-spaced data with a cadence of 29.4 minutes. lower left:

the time between observations. Missing measurements aside, the spacing between observations is

nearly uniform. the lower panel gives a closer look at the majority of the spacings, which are not

exactly the same but rather span a range of ±50ms around the 29.4-minute period observed in the

window function. lower right: the data power spectrum, showing approximate aliasing and clearly

indicating a peak near a period of 13.6 hours, along with higher-order components at multiples of

this value.

data in Figure 14 and Kepler data in Figure 15, are quite different, but in both cases essential

features of the observed power spectra can be understood by recognizing that the periodogram

measures not the power spectrum of the underlying signal, but a power specrtrum from the convo-

lution of the true signal transform and the Fourier transform of the window function.

5. From Classical to Lomb-Scargle Periodograms

Up until now, we have been mainly discussing the direct extension of the classical periodogram

in Equation 26 to non-uniform data. Returning to this definition, we can rewrite the expression in

a more suggestive way:

P (f) =1

N

∣∣∣∣∣N∑n=1

gne−2πiftn

∣∣∣∣∣2

=1

N

(∑n

gn cos(2πftn)

)2

+

(∑n

gn sin(2πftn)

)2 (31)

– 25 –

Although this form of the non-uniform periodogram can be useful for identifying periodic signals,

its statistical properties are not as straightforward as in the uniform case. When the classical peri-

odogram is applied to uniformly-sampled Gaussian noise, the values of the resulting periodogram

is chi-square distributed. This property becomes quite useful in practice when the periodogram is

used in the context of a classical hypothesis test to distinguish between periodic and non-periodic

objects—see Section 7.4.2. Unfortunately, when the sampling becomes nonuniform these properties

no longer hold and the periodogram distribution cannot in general be analytically expressed.

Scargle (1982) addressed this by considering a generalized form of the periodogram,

P (f) =A2

2

(∑n

gn cos(2πf [tn − τ ])

)2

+B2

2

(∑n

gn sin(2πf [tn − τ ])

)2

, (32)

where A, B, and τ are arbitrary functions of the frequency f and observing times {ti} (but not the

values {gn}), and showed that you can choose a unique form of A, B, and τ such that

1. The periodogram reduces to the classical form in the case of equally-spaced observations,

2. The periodogram’s statistics are analytically computable,

3. The periodogram is insensitive to global time-shifts in the data.

The values of A and B leading to these properties result in the following form of the generalized

periodogram:

PLS(f) =1

2

{ (∑n gn cos(2πf [tn − τ ])

)2/∑n cos2(2πf [tn − τ ])

+

(∑n gn sin(2πf [tn − τ ])

)2/∑n sin2(2πf [tn − τ ])

}(33)

where τ is specified for each f to ensure time-shift invariance:

τ =1

4πftan−1

(∑n sin(4πftn)∑n cos(4πftn)

). (34)

This modified periodogram differs from the classical periodogram only to the extent that the de-

nominators∑

n sin2(2πftn) and∑

n cos2(2πftn) differ from N/2, which is the expected value of

each of these quantities in the limit of complete phase sampling at each frequency. Thus, in many

cases of interest the Lomb-Scargle periodogram only differs slightly from the classical/Schuster

periodogram; an example of this is seen in Figure 16.

A remarkable feature of Scargle’s modified periodogram is that it is identical to the result

obtained by fitting a model consisting of a simple sinusoid to the data at each frequency f and con-

structing a “periodogram” from the χ2 goodness-of-fit at each frequency—an estimator which was

– 26 –

0.0

2.5

5.0

7.5

10.0

Spe

ctra

l Pow

er

Classical PeriodogramLomb-Scargle Periodogram

0.0 0.1 0.2 0.3 0.4 0.5frequency

0

5

10

15

20

25

30

ncos2[2 f(tn )]

nsin2[2 f(tn )]

Fig. 16.— upper panel: A comparison of the Classical periodogram (Equation 31) and the Lomb-

Scargle periodogram (Equation 33) for 30 noisy points drawn from a sinusoid. Though the two

periodogram estimates differ quantitatively, the essential qualitative features—namely the position

of significant peaks—typically remain the same. lower panel: the values of the denominators in

Equation 33. The difference between the Lomb-Scargle periodogram and the classical periodogram

stems from the difference between these quantities and N/2 = 15 (the dotted line).

considered in some depth by Lomb (1976). From this perspective, the τ shift defined in Equation 34

serves to orthogonalize the normal equations used in the least squares analysis. Partly due to this

deep connection between Fourier analysis and least-squares analysis, the modified periodogram in

Equation 33 has since become commonly referred to as the Lomb-Scargle Periodogram, although

versions of this approach had been employed even earlier (see, e.g. Gottlieb et al. 1975).

Because of the close similarity between the classical and Lomb-Scargle periodograms, the

bulk of or previous discussion applies—at least qualitatively—to periodograms computed with the

Lomb-Scargle method. In particular, reasoning about the effect of window functions on the observed

Lomb-Scargle power spectrum remains qualitatively useful even if it is not quantitatively precise.

One important caveat of the simple Lomb-Scargle formula is that the statistical guarantees

only apply when the observations have uncorrelated white noise; data with more complicated noise

characteristics must be treated more carefully; see, e.g., Vio et al. (2010) or the Least Squares

approach discussed in Section 6.1.

– 27 –

6. The Least-Square Periodogram and its Extensions

The equivalence of the Fourier interpretation and least squares interpretation of the Lomb-

Scargle periodogram allows for some interesting and useful extensions, some of which we will explore

in this section. First, let’s consider the Least Squares periodogram itself.

In the least squares interpretation of the periodogram, a sinusoidal model is proposed at each

candidate frequency f :

y(t; f) = Af sin(2πf(t− φf )) (35)

where the amplitude Af and phase φf can vary as a function of frequency. These model parameters

are fit to the data in the standard least-squares sense, by constructing the χ2 statistic at each

frequency:

χ2(f) ≡∑n

(yn − y(tn; f)

)2(36)

We can find the “best” model y(t; f) by minimizing χ2(f) at each freuqnecy with respect to Afand φf ; we will denote this minimum value as χ2(f). Scargle (1982) showed that with this setup,

the Lomb-Scargle periodogram from Equation 33 can be equivalently written:

P (f) =1

2

[χ2

0 − χ2(f)]

(37)

where χ20 is the non-varying reference model. The key realization here is that the Lomb-Scargle

periodogram essentially assumes a sinusoidal model for the data; this is visualized in Figure 17

for the data we had seen in Figure 1. This immediately begs the question: can we compute a

“periodogram” based on more general forms of the above model to more effectively fit the data?

0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50phase

15.6

15.8

16.0

16.2

mag

Fig. 17.— The sinusoidal model implied by the Lomb-Scargle periodogram for the LINEAR data

seen in Figure 1. Although a sinusoid does not perfectly fit the data, the sinusoidal model is close

enough that it serves to locate the correct frequency.

– 28 –

6.1. Measurement Errors

Perhaps the most important modification to the periodogram is to construct it such that it

correctly handles measurement error in the data. This can be done through the standard change

to the χ2 expression: i.e., if there are Gaussian errors σn on each observed yn, we can re-write

Equation 36 in the following (standard) way:

χ2(f) ≡∑n

(yn − ymodel(tn; f)

σn

)2

(38)

The periodogram constructed from this χ2 definition will more accurately reflect the spectral power

of noisy observations. The effect of this modification on Equation 33 is the addition of a multiplica-

tive weight 1/σn within each of the summations. Early versions of this sort of modification appeared

in Gilliland & Baliunas (1987) and Irwin et al. (1989); Scargle (1989) derived this “weighted” form

of the periodogram without direct reference to the least squares model, and Zechmeister & Kurster

(2009) showed that such a modification does not change the statistical properties of the resulting

periodogram.

This generalization of χ2(f) also suggests a convenient way to construct a periodogram in the

presence of correlated observational noise. If we let Σ denote the N × N noise covariance matrix

for N observations, and construct the vectors

~y = [y1, y2, · · · yn]T

~ymodel = [ymodel(t1), ymodel(t1), · · · ymodel(tn)]T , (39)

then the χ2 expression for correlated errors can be written

χ2(f) = (~y − ~ymodel)TΣ−1(~y − ~ymodel), (40)

which reduces to Equation 38 if noise is uncorrelated (i.e., if the off-diagonal terms of Σ are zero).

This resulting periodogram is quite similar to the approach to correlated noise developed by Vio

et al. (2010) from the Fourier perspective, and is in fact exactly equivalent in the case of the

“zero-mean colored noise” example considered therein.

6.2. Data Centering and the Floating Mean Periodogram

Another commonly-applied modification of the periodogram has variously been called the Date-

compensated Discrete Fourier Transform (Ferraz-Mello 1981), the floating-mean periodogram (Cum-

ming et al. 1999; VanderPlas & Ivezic 2015), or the generalized Lomb-Scargle Method (Zechmeister

– 29 –

& Kurster 2009), and involves adding an offset term to the sinusoidal model at each frequency3:

ymodel(t; f) = y0(f) +Af sin(2πf(t− φf )) (41)

This turns out to be quite important in practice, because the standard Lomb-Scargle approach

assumes that the data are pre-centered around the mean value of the (unknown) signal. In many

analyses, this requirement is accomplished by pre-centering data about the sample mean: this

approach is generally robust when the data provide full phase coverage of the observed signal; how-

ever, due to selection effects and survey cadence, full phase coverage can not always be guaranteed.

Using the sample mean in such cases can potentially lead to suppression of peaks of interest.

0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50phase

12

13

14

15

16

17

18

mag

floating mean modelstandard model

0.0

0.5

1.0 Floating Mean Periodogram

0.0 0.2 0.4 0.6 0.8 1.0Frequency

0.0

0.5

1.0 Standard Periodogram

Lom

b-S

carg

le P

ower

Fig. 18.— A comparison of the standard and floating mean periodograms for data with a frequency

of 0.3 and a selection effect which removes faint observations. In this case the mean estimated

from the observed data is not close to the true mean, which leads to the failure of the standard

periodogram to recover the correct frequency. A floating mean model correctly recovers the true

frequency of 0.3.

A simulated example of this is shown in Figure 18: the data consist of noisy observations of

a sinusoidal signal in which the faintest observations are omitted from the dataset (due to, e.g.,

a detection threshold). Applying the standard Lomb-Scargle periodogram to pre-centered data

leads to a periodogram that suppresses the true period of 0.3 days (lower right panel). Using the

floating-mean model of Equation 41 yields a periodogram that identifies this true period (upper

right panel). A detailed study of the floating-mean model is given by Zechmeister & Kurster

(2009) who show that the addition of the floating mean term does not change the useful statistical

properties outlined in Section 5.

3We choose to follow Cumming et al. (1999) and VanderPlas & Ivezic (2015) and call this a “Floating Mean”

model, to avoid confusion of the different uses of the term “generalized periodogram” in, e.g. Bretthorst (2001) and

Zechmeister & Kurster (2009).

– 30 –

6.3. Higher-order Fourier Models

A further generalization of the least squares periodogram involves multi-term Fourier models:

rather than fitting just a single sinusoid at each frequency, we might fit a partial Fourier series,

adding K − 1 additional sinusoidal components at integer multiples of the fundamental frequency:

ymodel(t; f) = A0f +

K∑k=1

A(k)f sin(2πkf(t− φ(k)

f )). (42)

Bretthorst (1988) takes a comprehensive look at this type of multi-term extension to the peri-

odogram, as well as related extensions to decaying signals and other more complex models.

14.5

15.0

P = 8.76 hours

1-term Model

0.0

0.5

1.01-term Periodogram

0.0 0.2 0.4 0.6 0.8 1.0phase

14.5

15.0

P = 17.52 hours

6-term Model

5.0 7.5 10.0 12.5 15.0 17.5 20.0period (hours)

0.0

0.5

1.06-term Periodogram

Fig. 19.— 1-term and 6-term Lomb-Scargle models fit to LINEAR object 14752041, an eclipsing

binary. Notice that the standard periodogram (upper panels) finds an alias of the true 17.5-hour

frequency, because a simple sinusoidal model cannot closely fit both the primary and secondary

eclipse. A six-term Fourier model, on the other hand, does find the true period (lower panels).

This kind of Fourier series generalization to the periodogram is quite tempting, because it

means that the periodogram can be tuned to fit models that are more complicated than simple

sine waves. In some cases, this can be very useful; for example, Figure 19 shows a Lomb-Scargle

analysis of an eclipsing binary star, characterized by both a primary and secondary eclipse. The

standard Lomb-Scargle periodogram (upper panels) is maximized at twice the true rotation fre-

quency, because the simple sinusoidal model is unable to closely fit the primary and secondary

eclipses separately. A six-term Fourier model (lower panels) is sufficiently flexible that it can detect

the true 17.5-hour period, though this comes at the expense of a much noisier periodogram.

This additional periodogram noise is easy to understand: in the least-squares view of the

periodogram, the periodogram height at any frequency is directly related to how well the model fits

the data. For nested linear models, adding additional terms will always provide an equal or better

fit to data than the simpler model, and so it follows that a periodogram based on a more complex

model will be higher at all frequencies, not just at frequencies of interest! Indeed, this effect can

be readily observed in Figure 19.

– 31 –

0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50phase

10

11

12

13

14

15

16

SA

P fl

ux (1

06 e/s

)

n=1n=2n=3

n=1

n=2

0 1 2 3 4 5 6 7 8Frequency (days 1)

n=3

Multi-term Periodograms for Kepler object ID 007198959

Fig. 20.— Multi-term models (left) and their corresponding periodograms (right) for the kepler

data shown in Figure 15

This background noise in the multi-term periodogram can also be understood in terms of

aliasing. Consider Figure 20, which shows several multi-term fits to the Kepler data from Figure 15.

Given a standard periodogram with a peak or sub-peak at f0, a 2-term periodogram will duplicate

this peak at f0/2, with the second harmonic driving the fit. Similarly, a 3-term periodogram will

add additional peaks both at f0/2 and f0/3, due to the original peak falling in the second and third

harmonic. In general, you can expect an N -term periodogram to contain N aliases of every feature

present in the standard periodogram; any strong peak revealed by the multi-term periodogram is

due to two or more of these aliases coinciding.

0.50 0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50phase

15.5

15.6

15.7

15.8

15.9

16.0

16.1

16.2

mag

n=1n=2n=3

n=1

n=2

2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5Frequency (days 1)

n=3

Multi-term Periodograms for LINEAR object ID 11375941

Fig. 21.— Multi-term models (left) and their corresponding periodograms (right) for the LINEAR

data shown in Figure 1

In cases where the single-term power spectrum is itself noisy and/or dominated by partial

aliasing due to window effects, these additional aliases can quickly wash-out any gains from the

more complicated model. For example, consider the multi-term periodograms for the LINEAR

data depicted in Figure 21: as previously discussed, the LINEAR periodogram contains a number

of aliases of the true peak due to the dominant 1-day signal in the window function. Adding terms

– 32 –

to this model only compounds this problem, increasing the number of spurious peaks at the low-

frequency end. For data with even moderate levels of noise, chance coincidences of these peaks

can lead to spurious detections which dominate the true peak, particularly for models with many

Fourier terms.

6.4. Additional Extensions

When the periodogram is viewed from the least-squares model-fitting perspective, there is no

need to limit the analysis to sums of sinusoids. There have been some very interesting applications

extending this type of analysis to more arbitrary models. For example, periodogram models can be

extended to measure decaying signals, non-stationary signals, multi-frequency signals, chirps, and

other signal types (see, e.g. Jaynes 1987; Bretthorst 1988; Gregory 2001). In particular, Bretthorst

(1988) demonstrates the effectiveness of such approaches in applications ranging from medical

imaging to astronomy. The challenge of such extensions is the fact that you often need to use

some prior knowledge of the system being observed to decide whether a more complicated model is

indicated, as well as what form of model to apply. In practice this comes up only when searching

for very specific signals for which a more complicated model has some a priori physical motivation.

On the astronomy side, several examples of this flavor of extension exist. For example, the

Supersmoother approach to detecting periodicity involves fitting a flexible non-parametric smooth-

ing function to the data at each frequency (Reimann 1994): the flexibility of the model leads to

fewer aliasing issues when compared to the more constrained sinusoidal model. Another approach

is to use empirically-derived templates as a functional fit at each frequency; this has been employed

effectively in Sesar et al. (2010, 2013) and related work.

Finally, there has been some exploration of extensions to the Lomb-Scargle periodogram for

use with multi-band observations, using various forms of regularization to control model complexity

(VanderPlas & Ivezic 2015; Long et al. 2016). With many of these least-squares and/or Bayesian ex-

tensions, computational complexity quickly becomes an issue, because fast O(N logN) approaches

which can be used for sinusoidal fits (see Section 7.6) are not available for more general functional

forms, though there is some promising work in this area: see, for example, the Fast Template Pe-

riodogram4 (Hoffman et al. 2017, in prep) which can quickly construct periodograms from Fourier

approximations to templates.

4http://ascl.net/code/v/1559

– 33 –

6.5. A Note About “Bayesian” Periodograms

The least squares view of the Lomb-Scargle periodogram creates a natural bridge, via maximum

likelihood, to Bayesian periodic analysis. In fact, Jaynes (1987) showed that the standard form

of the Lomb-Scargle periodogram can be derived directly from the axioms of Bayesian probability

theory outlined in his comprehensive treatment of the subject (Jaynes & Bretthorst 2003)5. In

the Bayesian view, the Lomb-Scargle periodogram is in fact the optimal statistic for detecting a

stationary sinusoidal signal in the presence of Gaussian noise.

For the standard, simple-sinusoid model, the Bayesian periodogram is given by the posterior

probability of frequency f given the data D and sinusoidal model M :

p(f | D,M) ∝ ePLS(f) (43)

where PLS(f) is the Lomb-Scargle power from Equation 33. The effect of this exponentiation, as

seen in Figure 22, is to suppress side-lobes and alias peaks in relation to the largest one of the

spectrum.

0 10 20 30 40 50time

1.5

1.0

0.5

0.0

0.5

1.0

1.5

sign

al

Data and Model

0.00

0.25

0.50

P LS

Lomb-Scargle Periodogram

0.0 0.2 0.4 0.6 0.8 1.0frequency

0

50

p(f|D

)

Bayesian Periodogram

Fig. 22.— Comparison of the Lomb-Scargle periodogram (upper right) and the Bayesian posterior

periodogram (lower right) for simulated data (left) drawn from a simple sinusoid. The Bayesian

posterior is equal to the exponentiation of the unnormalized Lomb-Scargle periodogram, and thus

tends to suppress all but the largest peak. The Bayesian approach can be useful, but is problematic

if not used carefully (see text for discussion).

The benefit of the Bayesian approach is that it allows more flexible models, more principled

treatment of nuisance parameters related to noise in the data, and the ability to make use of

prior information in a periodic analysis. It explicitly returns probability density as a function of

frequency, and it is tempting to think of this as the “natural” way to interpret the periodogram.

The problem, however, is that the Bayesian periodogram does not compute the probability

that the data are periodic with a given frequency, but rather that probability conditioned on a

5Read this. Really.

– 34 –

relatively strong assumption: that the data are drawn from a sinusoidal model. As such, the

standard Bayesian periodogram is not a useful measure of generic periodocity! In fact, the Baysian

periodogram’s derivation under the assumption of a sinusoidal signal is perhaps the best argument

against its use for unknown signals: the result of a Bayesian analysis is only ever as good as the

assumptions that go into it, and for a general (not-necessarily sinusoidal) signal, those assumptions

are incorrect, and the periodogram should not be trusted.

Now, the standard Lomb-Scargle periodogram can also be viewed as derived from a sinusoidal

fit, and thus might appear subject to the same criticism. But unlike the Bayesian periodogram,

the standard periodogram affords interpretation in light of Fourier analysis and window functions:

here the incorrect sinusoidal model manifests itself in terms of frequency aliases, which can be

understood through the analysis of window functions. In most cases of interest, such analysis turns

out to be vital to the application and interpretation of the periodogram (see Section 7.2).

In other words, the Bayesian approach essentially goes “all-in” on the least squares interpreta-

tion of the periodogram, exponentially suppressing the information that allows you to reason about

the periodogram in light of its connection to Fourier analysis. In contrived cases with clean sinu-

soidal data and unstructured window functions, exponential attenuation of side-lobes and aliases

may seem appealing (see, e.g. Mortier et al. 2015), but that appeal extends to the real world only

in the most favorable of cases—i.e., high signal-to-noise measurements of near-sinusoidal data with

a very well-behaved survey window. In short, you shiould be wary of placing too much trust in a

Bayesian periodogram, unless you’re certain of the type of signal you’re looking for.

This is not to say that all Bayesian approaches to periodic analysis are similarly flawed; there

have been many interesting studies that go beyond the simple sinusoidal model and use more

complex and/or flexible models. Some examples are models based on Fourier extensions with

strong priors (e.g. Bretthorst 1988), instrument-dependent parametric models (e.g. Angus et al.

2016), flexible non-parametric functions (e.g. Gregory & Loredo 1992), Gaussian Process models

(e.g. Wang et al. 2012), and specially-designed stochastic models (e.g. Kelly et al. 2014). Though

these are often be more accurate and powerful than the Lomb-Scargle approach, they tend to be

far more expensive computationally. Bayesian approaches based on Markov Chain Monte Carlo

(MCMC) also tend to run into stability problems, particularly for multimodal or other complicated

posterior distributions (See, for example, the RR Lyrae discussion in Kelly et al. 2014).

7. Practical Considerations when using Lomb-Scargle Periodograms

The previous sections have given a conceptual introduction to the Lomb-Scargle periodogram

and its roots in both Fourier and Least-squares analysis. This section gets to the meat of the subject

at hand: given this understanding of the Lomb-Scargle periodogram and related approaches, how

should we use it most effectively in practice? The following sections will identify several of the

important issues and questions that are not often addressed in the literature on the subject, but

– 35 –

are nevertheless vital to consider when using the periodogram in practice.

7.1. Choosing a Frequency Grid

The frequency grid used for a Lomb-Scargle analysis is an important choice that is too-

often glossed-over, probably because the choice is so straightforward in the more familiar case

of uniformly-sampled data. For non-uniform data, it is not so simple, and there are two important

considerations: the frequency limits and the grid spacing.

The frequency limit on the low end is relatively easy: for a set of observations spanning a length

of time T , a signal with frequency 1/T will complete exactly one oscillation cycle, and so this is

a suitable minimum frequency. Often, it’s more convenient just to set this minimum frequency

to zero, as it doesn’t add much of a computational burden and is unlikely to add any significant

spurious peak to the periodogram.

The high-frequency limit is more interesting, and goes back to the discussion of Nyquist and/or

limiting frequencies from Section 4.1: in order to not miss relevant information, it is important to

compute the periodogram up to some well-motivated limiting frequency fmax. This could be a

true Nyquist limit based on the Eyer & Bartholdi (1999) definition, a pseudo-Nyquist limit based

on careful scrutiny of the window function (cf. Figure 15), a limiting frequency based on the

integration time of individual observations, or a limit based on prior knowledge of the kinds of

signals you expect to detect.

1 2 3 4 5 6period (hours)

0.0

0.2

0.4

0.6

Lom

b-S

carg

le P

ower

LINEAR object 11375941

Well-motivated frequency grid10,000 equally-spaced periods

Fig. 23.— An example of a poorly-chosen frequency grid for the data in Figure 1. The dark curve

shows a periodogram evaluated on a grid of 10,000 equally-spaced periods; the light curve shows

the true periodogram (evaluated at ∼200, 000 equally-spaced frequencies). This demonstrates that

using too coarse a grid can lead to a periodogram that entirely misses relevant peaks.

With the frequency range decided, we next must determine how finely to sample the frequencies

between the limits. This choice turns out to be quite important as well: too fine a grid can lead

to unnecessarily long computation times that can add up quickly in the case of large surveys,

– 36 –

while too coarse a grid risks entirely missing narrow peaks that fall between grid points. For

example, Figure 23 shows the true, well-sampled periodogram (gray line), along with a periodogram

computed for 10,000 equally-spaced periods covering the same range (black line). Because the

spacing of the 10,000-point grid is much larger than the width of the periodogram peaks, the

analysis entirely misses the most important peaks in the periodogram!

This shows us that it’s important to choose grid spacings smaller than the expected widths of

the periodogram peaks. From our discussion of windowing in Section 3 (particularly Figure 6), we

know that data observed through a rectangular window of length T will have sinc-shaped peaks of

width ∼1/T . To ensure that our grid sufficiently samples each peak, it is prudent to over-sample

by some factor—say no samples per peak—and use a grid of size ∆f = 1noT

. This pushes the total

number of required periodogram evaluations to

Neval = noTfmax (44)

So what is a good choice for no? Values ranging from no = 5 (Schwarzenberg-Czerny 1996; Vander-

Plas & Ivezic 2015) to no = 10 (Debosscher et al. 2007; Richards et al. 2012) seem to be common

in the literature; for periodograms computed in this paper, we use no = 5.

Depending on the characteristics of the dataset, the size of the resulting frequency grid can

vary greatly. For example, the Kepler data shown in Figure 15 has a pseudo-Nyquist frequency

of 48.9 days−1 and an observing window of T = 90 days. To compute five samples per peak thus

requires Neval ≈ 22, 000 evaluations of the periodogram. On the other hand, the LINEAR data

shown in Figure 1 does not have any noteable aliasing structure in its window function. In this case

the maximum detectable frequency is the Nyquist limit defined by its temporal resolution, which is

6 digits beyond the decimal point in days. From Equation 30, we can write fNy = 500, 000 days−1,

and given the observing window of T = 1962 days, we find that five evaluations per peak across the

entire detectable frequency range would require Neval ≈ 4.9× 109 evaluations of the periodogram!

Computing this large a periodogram in most cases is computationally intractable (see Section 7.6),

and so in practice one must choose a smaller limiting frequency based on prior information about

what kind of signals are expected in the data: for example, in Figure 2, we chose a limiting frequency

of fmax = (1 hour)−1 based on typical oscillation periods expected for SX Phe-type stars. This

leads to a much more manageable Neval ≈ 240, 000 periodogram evaluations.

By comparison, data from the LSST survey (Ivezic et al. 2008) will fall somewhere in-between:

full frequency coverage of the 10-year data up to a limiting frequency defined by the 30-second

integration time for each visit would require roughly 25 million periodogram evaluations per object,

which for fast implementations (see the next section) could be accomplished in several seconds on

a modern CPU.

One final note: although it can be more easily interpretable to visualize periodograms as a

function of period rather than a function of frequency, the peak widths are not constant in period.

Regular grids in period rather than frequency tend to over-sample at large periods and under-sample

at small periods; for this reason it is preferable to use a regular grid in frequency.

– 37 –

7.2. Failure Modes

When using the Lomb-Scargle periodogram in an observational pipeline, it is vital to keep

in mind the failure modes of the periodogram approach, which are rooted in the aliasing and

pseudo-aliasing effects rooted in the structure of the window function (recall Section 3). Due to

the interaction of the signal, the convolution due to the survey window, and noise in the data, it

is quite common for the largest peak in the periodogram to correspond not to the true frequency,

but some alias of that frequency.

0.0 0.2 0.4 0.6 0.8 1.0 1.2True Period

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

Per

iodo

gram

Pea

k

363 / 1000(a) 158 / 1000(b)

49 / 1000(c) 13 / 1000(d)

Fig. 24.— Comparison of the true period and peak Lomb-Scargle period for 1000 simulated periodic

light curves. Each has 60 irregular observations over 180 nights, with a cadence typical of ground-

based surveys (i.e., showing a strong diurnal window pattern similar to Figure 13). The Lomb-

Scargle peak does not always coincide with the true period, and there is noticeable structure among

these failures. Panels (a)-(d) isolate some of the specific modes of failure that should be expected

for this kind of window function; see the text for more discussion.

Figure 24 demonstrates this for some simulated data. The data consist of 60 noisy observations

each of 1,000 simulated light curves within a span of 180 days. The window is typical of ground-

based data, with each observation recorded within a few hours before or after midnight each night.

The left panel shows the results: the peridogram peak coincides with the true period in under 50%

of cases, and the modes of failure lead to noticeable patterns in the resulting plot. Though these

are simulated observations, the pattern seen here is typical of real observations: see, for example,

VanderPlas & Ivezic (2015) and Long et al. (2016) which show similar plots for RR Lyrae candidates

from the Sloan Digital Sky Survey.

– 38 –

These patterns can be understood in terms of the interaction between the window function and

the underlying spectral power. As we discussed in Section 3, a nightly observation pattern—typical

of ground-based surveys—leads to a window function with a strong diurnal component that causes

each frequency signature f0 to be partially aliased at f0 +nδf , for integers n and δf = 1 cycle/day.

(Recall Figures 13-14). This is the source of the failure modes depicted in panel (a) of Figure 24;

in terms of periods the above expression becomes:

Pobs =

(1

Ptrue+ n δf

)−1

(45)

where, to be clear, n is a positive or negative integer and δf is the frequency of a strong feature

in the window (here, 1 cycle/day). For the simulated data, close to 36% of the objects are mis-

characterized along these failure modes.

Another mode of failure is when the periodogram for an object with frequency f0 isolates a

higher harmonic of that fundamental frequency, such as 2f0. This may occur for periodic signals

that are not strictly sinusoidal, and so have power at a higher harmonics. This higher harmonic

peak is also subject to the same aliasing effects as in Equation 45. Thus we can extend Equation 45

and describe the failure modes depicted in panel (b) of Figure 24 with the following equation, for

positive integers m > 0:

Pobs =

(m

Ptrue+ nδf

)−1

(46)

Panel (b) of Figure 24 illustrates this for m = 2; this harmonic and its aliases account for roughly

15% of the results.

Panel (c) of Figure 24 shows the final pattern of failure in Lomb-Scargle results, which has an

opposite trend with true period. This is closely related to the effects shown in panels (a) and (b),

but comes from the even symmetry of the Lomb-Scargle periodogram. Every periodogram peak at

frequency f0 has a corresponding peak at −f0, and this is true of aliases as well. If a peak’s aliases

cross into the negative frequency regime, they are effectively reflected into the positive-frequency

range. This reflection can be accounted for by a further modification of Equation 46—taking its

absolute value:

Pobs =

∣∣∣∣ m

Ptrue+ nδf

∣∣∣∣−1

(47)

Panel (c) shows the 5% of objects that fall along this reflected failure mode for m = 1, n = −2.

After accounting for all these known sources of periodogram failure, only roughly 1% of points are

misclassified in an “unexplained” way, seen in panel (d).

When applying a periodogram in practice, it is vital to take such effects into account, rather

than blindly relying on the single periodogram peak as your best estimate of the period. Applying

understanding of windowing and aliasing effects can help in detecting failures of the periodogram,

but is no silver bullet. For an observed peak at fpeak from a survey whose window has strong power

at δf , something like the following should probably be employed:

– 39 –

1. Check for a peak at fpeak/m for at least m ∈ {2, 3}. If a significant peak is found, then fpeakis probably an order-m harmonic of the true frequency.6

2. Check for peaks at |fpeak ±nδf | for at least n ∈ {1, 2} (where δf is determined from plotting

the survey window power, and is generally (1 day)−1 for ground-based surveys). If these

aliases exist, then it is possible that you have found a peak on the sequence of expected

aliases—though keep in mind that there is no way to know from the periodogram alone

whether or not this is the “true” peak!

3. For each of the top few of these aliases, fit a more complicated model (such as a multi-term

Fourier series, template-based fit, etc.) to select among them.

For noisy observations, this procedure cannot generally guarantee that you have found the correct

peak, but it is far preferable to the simplistic approach of blindly trusting the highest peak in

the periodogram! We will come back to the question of uncertainty in the periodogram result in

Section 7.4.

7.3. Window Functions and Deconvolution

Our discussion of window functions here and in Section 3 has highlighted the impact of struc-

ture in the survey window on the resulting observed power spectrum, and the importance of ex-

amining that window power to understand features of the resulting periodogram. Here we will

consider the computation of the window function, as well as the possibility of recovering the true

periodogram by deconvolving the window.

7.3.1. Computing the Window Function

The window power spectrum can be computed directly from the delta-function representation

of the window function; From Equation 1, Equation 9, and Equation 27, we can write

PW (f ; {tn}) =

∣∣∣∣∣N∑n=1

e−2πiftn

∣∣∣∣∣2

(48)

Comparing this to Equation 26, we see that this is essentially the classical periodogram for data

gn = 1 at all times tn. With this fact in mind, one convenient way to estimate the form of the window

power is to compute a standard Lomb-Scargle periodogram on a series of unit measurements,

making sure to not pre-center the data or to use a floating mean model. As Scargle (1982) notes,

6Notice that this step can detect aliases like those encoundered in Figure 19, without resorting to a problematic

multiterm model.

– 40 –

this computation does not give the true window—it differs from the true window just as the Lomb-

Scargle periodogram differs from the classical periodogram—but in practice it is accurate enough

to give useful insight into the window function’s important features. This method is how window

power spectra has been computed throughout this paper.

7.3.2. Deconvolution and CLEANing

With this ability to compute the window function, we might hope to be able to use it quan-

titatively to remove spurious peaks from the observed power spectrum. Recall from Equation 28

that the observed data are a point-wise product of the underlying function and the survey window:

gobs(t) = g(t)W{tn}(t), (49)

and the convolution theorem tells us that the observed Fourier transform is a convolution of the

true transform and the window transform:

F{gobs} = F{g} ∗ F{W{tn}} (50)

Given this relationship, we might hope to be able to invert this convolution to recover F{g} directly.

For example, we could write:

g(t) = gobs(t)/W{tn}(t)

F{g} = F{gobs/W{tn}}= F{gobs} ∗ F{1/W{tn}} (51)

Because of the localization of observations, W{tn}(t) is zero for most values of t and so 1/W{tn}(t)

and its Fourier transform are not well defined. Because of this, direct deconvolution is not an option

in most cases of interest—in other words, the deconvolution problem is ill-posed and does not have

a unique solution.

There have been a few attempts in the literature to use procedural algorithms to solve this

under-constrained deconvolution problem, perhaps most notably by adapting the iterative CLEAN

algorithm developed for deconvolution in the context of radio astronomy (Roberts et al. 1987). For

cleaning of Lomb-Scargle periodograms, the CLEAN approach is hindered by three main deficien-

cies: first and most importantly, the CLEAN algorithm at each iteration assumes that the highest

peak is the location of the primary signal; this is not always borne out for realistic observations of

faint objects where the cleaning is most necessary (recall the discussion in Section 7.2). Second,

the convolution takes place at the level of the Fourier transform rather than the PSD; trying to

clean a PSD directly ignores important phase information. Third, the CLEAN algorithm assumes a

classical FFT analysis: while the Lomb-Scargle periodogram is equivalent to classical periodogram

in the limit of equally-spaced observations, it is not equivalent in the relevant case of unequal ob-

servations (see Section 5), and so any attempt to apply CLEAN to Lomb-Scargle analysis directly

would be fundamentally flawed even if it were not ill-posed to begin with.

– 41 –

The latter two issues could be remedied by focusing on the non-uniform Fourier transform

and the classical periodogram rather than the Lomb-Scargle modification, but doing so would

jettison the benefits of Lomb-Scargle—i.e., its useful statistical properties and extensibility via

least squares—and this would still not address the far more problematic first issue. My feeling

is that there is room for a more principled approach to the unconstrained deconvolution problem

for periodograms derived from non-uniform fast Fourier transforms (perhaps through some sort of

L1/lasso regularization that imposes assumptions of sparsity on the true periodogram) but to date

this does not appear to have been explored anywhere in the literature.

7.4. Uncertainties in Periodogram Results

An important aspect of reporting results from the Lomb-Scargle periodogram is the uncertainty

of the estimated period. In many areas of science we are used to being able to report uncertainties

in terms of errorbars, e.g. “the period is 16.3 ± 0.6 hours”. For periods derived from the Lomb-

Scargle periodogram, however, uncertainties generally cannot be meaningfully expressed in this

way: as we saw in the discussion of failure modes in Section 7.2, the concern for periodograms

is more often a disjointed inaccuracy associated with false peaks and aliases, rather than a more

smooth imprecision in location of a particular peak.

7.4.1. Peak Width and Frequency Precision

A periodic signal will be reflected in the periodogram by the presence of a peak of a certain

width and height. In the Fourier view, the precision with which a peak’s frequency can be identified

is directly related to the width of this peak; often the half-width at half-maximum f1/2 ≈ 1/T is

used. This can be formalized more precisely in the least squares interpretation of the periodogram,

in which the inverse of the curvature of the peak is identified with the uncertainty (Ivezic et al.

2014)—which in the Bayesian view amounts to fitting a Gaussian curve to the (exponentiated) peak

(Jaynes 1987; Bretthorst 1988). This introduces first-order dependence on the number of samples

N and their average signal-to-noise ratio Σ; the scaling is approximately (see, e.g. Gregory 2001):

σf ≈ f1/2

√2

NΣ2. (52)

This dependence comes from the fact that the Bayesian uncertainty is related to the width of the

exponentiated periodogram, which depends on Pmax, the height of the peak7.

7 To see why, consider a periodogram with maximum value Pmax = P (fmax), so that P (fmax ± f1/2) = Pmax/2.

The Bayesian uncertainty comes from approximating the exponentiated peak as a Gaussian; i.e., exp[P (fmax±δf)] ∝exp[−δf2/(2σ2

f )]. From this we can write Pmax/2 ≈ Pmax − f21/2/(2σ

2f ) or σf ≈ f1/2/

√Pmax. In terms of signal-to-

noise ratio Σ = rms[(yn − µ)/σn], a well-fit model gives Pmax ≈ χ20/2 ≈ Σ2N/2, which leads to the expression in

Equation 52.

– 42 –

We have motivated from the Fourier window discussions that peak width f1/2 is the inverse

of the observational baseline; the surprising result is that to first order the peak width in the

periodogram does not depend on either the number of observations or their signal-to-noise ratio!

This can be seen visually in Figure 25, which shows periodograms for simulated data with a fixed

4-period baseline, with varying sample sizes and signal-to-noise ratios. In all cases, the widths

of the primary peak are essentially identical, regardless of the quality or quantity of data! Data

quality and quantity are reflected in the height of the peak in relation to the “background noise”,

which speaks to the peak significance rather than the precision of frequency detection.

0.00

0.25

0.50

0.75

1.00

P LS

(nor

mal

ized

)

Peak scaling with number of data points (fixed S/N=10)

N=1000N=100N=10

0.00

0.25

0.50

0.75

1.00

P LS

(nor

mal

ized

)

Peak scaling with signal-to-noise ratio (fixed N=1000)

S/N=10S/N=1S/N=0.1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0frequency

0

2500

5000

7500

10000

P LS/(

S/N

)2 (P

SD

-nor

mal

ized

)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0frequency

0

250

500

750

1000P L

S/(S/

N)2

(PS

D-n

orm

aliz

ed)

Fig. 25.— The effect of the number of points N and the signal-to-noise ratio S/N on the expected

width and height of the periodogram peak. Top panels show the normalized periodogram (Equa-

tion 59), while bottom panels show the PSD-normalized periodogram (Equation 33) scaled by noise

variance. Perhaps surprisingly, neither the number of points nor the signal-to-noise ratio affects

the peak width.

For this reason, if you insist on reporting frequency errorbars derived from peak width, the

results can be pretty silly for observations with long baselines. For example, going by the peak

width in Figure 2, the periodogram reveals a period of 2.58014±0.00004 hours, a relative precision

of about 1/1000 percent. While this is an accurate characterization of the period precision assuming

that peak is the correct one, it does not capture the much more relevant uncertainties demonstrated

in Section 7.2, in which we might find ourselves on the wrong peak entirely. This is why in general,

peak widths and Gaussian error bars should generally be avoided when reporting uncertainties in

the context of a periodogram analysis.

7.4.2. False Alarm Probability

A much more relevant quantity for expressing uncertainty of periodogram results is the height

of the peak, and in particular the height compared to the spurious background peaks that arise

in the periodogram. Figure 25 indicates that this property does depend on both the number of

– 43 –

observations and their signal-to-noise ratio: for the small-N/low signal-to-noise cases, the spurious

peaks in the background become comparable to the size of the true peak. In fact, as we saw in

Section 5, the ability to analytically define and quantify the relationship between peak height and

significance is one of the primary considerations that led to the standard form of the Lomb-Scargle

periodogram.

The typical approach to quantifying the significance of a peak is the False Alarm Probability

(FAP), which measures the probability that a dataset with no signal would—due to coincidental

alignment among the random errors—lead to a peak of a similar magnitude. Scargle (1982) showed

that for data consisting of pure Gaussian noise, the values of the unnormalized periodogram in

Equation 33 follows a χ2 distribution with two degrees of freedom; that is, at any given frequeny

f0, if Z = P (f0) is the periodogram value from Equation 33, then

Psingle(Z) = 1− exp(−Z) (53)

is the cumulative probability of observing a periodogram value less than Z, in data consisting only

of Gaussian noise.8

Independent Frequency Method

We are generally not interested in the distribution of one particular randomly-chosen frequency, but

rather the distribution of the highest peak of the periodogram, which is a quite different situation.

By analogy, consider rolling a standard 6-sided die. The probability, in a single roll, of rolling

a number, say r > 4 is easy to compute: it’s 2 sides out of six, or p(r > 4) = 1/3. If, on the other

hand, you roll 10 dice and choose the largest number among them, the probability that it will be

greater than 4 is far larger than 1/3; it is p(max10(r) > 4) = 1 − (1 − 1/3)10 ≈ 0.98. The case

for the periodogram is analogous: the probability of seeing a spurious peak at any single location

(Equation 53) is relatively small, but the probability of seeing a single spurious peak among a large

number of frequencies is much higher.

With the dice, this calculation is easy because the rolls are independent: the result of one roll

does not affect the result of the next. With the periodogram, though, the value at one frequency

is correlated with the value at other frequencies in a way that is quite difficult to analytically

express—these correlations come from the convolution with the survey window. Nonetheless, one

common approach to estimating the distribution has been to assume it can be modeled on some

“effective number” of independent frequencies Neff , so that the FAP can be estimated as

FAP (z) ≈ 1−[Psingle(z)

]Neff (54)

8 Be aware that for different periodogram normalizations (See Section 7.5), the form of this distribution changes;

see Cumming et al. (1999) or Baluev (2008) for a good summary of the statistical properties of various periodogram

normalizations.

– 44 –

A very simple estimate for Neff is based on our arguments of the expected peak width, δf = 1/T .

In this approximation, the number of independent peaks in a range 0 ≤ f ≤ fmax is assumed to

be Neff = fmaxT . There have been various attempts to estimate this value more carefully, both

analytically and via simulations (see, e.g. Horne & Baliunas 1986; Schwarzenberg-Czerny 1998;

Cumming 2004; Frescura et al. 2008), but all such approaches are necessarily only approximations.

Baluev (2008) Method

A more sophisticated treatment of the problem is that of Baluev (2008), who derived an analytic

result based on theory of extreme values for stochastic processes. For the standard periodogram

in Equation 33, Baluev (2008) showed that the following formula for the FAP provides a close

upper-limit even in the case of highly structured window functions:

FAP (z) ≈ 1− Psingle(z)e−τ(z) (55)

where, for the normalized periodogram (Equation 59),

τ(z) ≈W (1− z)(N−4)/2√z (56)

and W = fmax√

4π var(t) is an effective width of the observing window in units of the maximum

frequency chosen for the analysis. This result is not an exact measure of the FAP, but rather an

upper limit that is valid for alias-free periodograms, i.e. cases where the window function has very

little structure, but which holds quite well even in the case of more realistic survey windows.

Bootstrap Method

In the absence of a true analytic solution to the false alarm probability, we can turn to computational

methods such as the bootstrap. The bootstrap method is a technique in which the statistic in

question is computed repeatedly on many random resamplings of the data in order to approximate

the distribution of that statistic (see Ivezic et al. 2014, for a useful general discussion of this

technique). For the periodogram, in each resampling we keep the temporal coordinates the same,

draw observations randomly with replacement from the observed values, and then compute the

maximum of the resulting periodogram. For enough resamplings, the distribution of these maxima

will approximate the true distribution for the case with no periodic signal present. The bootstrap

produces the most robust estimate of the FAP because it makes few assumptions about the form

of the periodogram distribution, and fully accounts for survey window effects.

Unfortunately, the computational costs of the bootstrap can be quite prohibitive: to accurately

measure small levels of FAP requires constructing a large number of full periodograms. If you are

probing a false positive rate of r among N bootstrap reseamples, you would roughly expect to find

rN ±√rN relevant peaks in your bootstrap sample. More concretely, for 1,000 bootstrap samples,

you’ll find only ∼ 10± 3 peaks reflecting a 1% false positive rate. The random fluctuations in that

– 45 –

count translate directly to an inability to accurately estimate the false positive rate at that level.

A good rule-of-thumb is that to accurately characterize a false positive rate r requires something

like ∼ 10/r individual periodograms to be computed—this grows computationally intractable quite

quickly. The bootstrap is also not universally applicable: for example, it does not correctly account

for cases where noise in observations is correlated; for more discussion of the bootstrap approach

and its caveats, see Ivezic et al. (2014) and references therein.

0.00 0.05 0.10 0.15 0.20 0.25 0.30Observed Maximum Peak Value

10 3

10 2

10 1

100

Fals

e A

larm

Pro

babi

lity

100 observations over 5 days; fmax = 10 days 1

Bootstrap, unstructured windowBootstrap, structured windowBaluev methodapprox. Neff method

Fig. 26.— Comparison of various approaches to computing the false alarm probability (FAP) for

simulated observations with both structured and unstructured survey windows. Note that the

approximate Neff method in its most naıve form tends to underestimate the bootstrapped FAP,

while the Baluev (2008) method tends to overestimate the bootstrapped FAP, particularly for data

with a highly-structured survey window.

Figure 26 compares these methods of estimating the false alarm probability, for both an un-

structured survey window, and a structured window which produces the kinds of aliasing we have

discussed above. The naıve approximation in this case under-estimates the FAP at nearly all

levels—so, for example, it might lead you to think a peak has a FAP of 10% when in fact it is

closer to 30%. The Baluev method, by design, over-estimates the FAP, and is quite close to the

bootstraped distribution in the case of an unstructured window. For a highly structured window,

the Baluev method does not perform as well, but still tends to over-estimate the FAP—so, for

example, it might lead you to think a peak as a FAP of 10% when in fact it is closer to 5%.

– 46 –

Finally, I will note that Suveges (2012) has shown the promise of a hybrid approach of the

bootstrap method and the extreme value statistics of Baluev (2008); this has the ability to compute

accurate FAPs without the need to compute thousands of full bootstrapped periodograms.

What Does the False Alarm Probability Measure?

While the False Alarm Probability can be a useful concept, you should be careful to keep in mind

that it answers a very specific question: “what is the probability that a signal with no periodic

component would lead to a peak of this magnitude?”. In particular, it emphatically does not answer

the much more relevant question “what is the probability that this dataset is periodic given these

observations?”. This boils down to a case of P (A | B) 6= P (B | A), but still it is common to see

the false alarm probability treated as if it speaks to the second rather than the first question.

Similarly, the false alarm probability tells us nothing about the false negative rate; i.e. the

rate at which we would expect to miss a periodic signal that is, in fact, present. It also tells us

nothing about the error rate; i.e. the rate at which we would expect to identify an incorrect alias

of a signal present in our data (as we saw in Section 7.2). Too often users are tempted to interpret

the FAP too broadly, with the hope that it would speak to questions that are beyond its reach.

Unfortunately, there is no silver bullet for answering these broader, more relevant questions of

uncertainty of Lomb-Scargle results. Perhaps the most fruitful path toward understanding of such

effects for a particular set of observations—with particular noise characteristics and a particular

observing window—is via simulated data injected into the detection pipeline. In fact, this type of

simulation is a vital component of most (if not all) current and future time-domain surveys that

seek to detect, characterize, and catalog periodic objects, regardless of the particular approach used

to identify periodicity (see, e.g. Delgado et al. 2006; Ridgway et al. 2012; Ivezic et al. 2008; Sesar

et al. 2010; Oluseyi et al. 2012; McQuillan et al. 2012; Drake et al. 2014, etc.)

7.5. Periodogram Normalization and Interpretation

Often it is useful to be able to interpret the periodogram values themselves. Recall that there

are two equally valid perspectives of what the periodogram is measuring—the Fourier view and

the least-squares view—and each of these lends itself to a different approach to normalizion and

interpretation of the periodogram.

7.5.1. PSD normalization

When considering the periodogram from the Fourier perspective, it is useful to normalize the

periodogram such that in the special case of equal spaced data, it recovers the standard Fourier

power spectrum. This is the normalization used in Equation 33 and the equivalent least-squares

– 47 –

expression seen in Equation 37:

P (f) =1

2

[χ2

0 − χ2(f)]

(57)

For equally-spaced data, this periodogram is identical to the standard definition based on the fast

Fourier transform:

P (f) =1

N|FFT (yn)|2 (58)

in particular, this means that the units of the periodogram are unit(y)2, and can be interpreted

as squared-amplitudes of the Fourier component at each frequency. Note, however, that the units

change if data uncertainties are incorporated into the periodogram as in Section 6.1; the peri-

odogram in this normalization becomes unitless: it is essentially a measure of periodic content in

signal-to-noise ratio rather than in signal itself.

7.5.2. Least-squares normalization

In the least-squares view of the periodogram, the periodogram is interpreted as an inverse

measure of the goodness-of-fit for a model. When we express the periodogram as a function of χ2

model fits as in Equation 57, it becomes clear that the periodogram has a mathematical maximum

value: if the sinusoidal model perfectly fits the data at some frequency f0, then χ2(f0) = 0 and the

periodogram is maximized at a value of χ20/2. On the other end, it is mathematically impossible

for a best-fit sinusoidal model at any frequency to do worse than the simple constant, non-varying

fit reflected in χ20, and so the minimum value of Equation 57 is exactly zero.

This suggests a different normalization of the periodogram, where the values are dimensionless

and lie between 0 and 1:

Pnorm(f) = 1− χ2(f)

χ20

(59)

This “normalized periodogram” is unitless, and is directly proportional to the unnormalized (or

PSD-normalized) periodogram in Equation 57.

While the normalization does not affect the shape of the periodogram, its main practical

consequence is that the statistics of the resulting periodogram differ slightly, and this needs to be

taken into account when computing analytic estimates of uncertainties and false alarm probabilities

explored in Section 7.4.2. Other normalizations exist as well, but seem to be rarely used in practice.

For a concise summary of several periodogram normalizations and their statistical properties, refer

to the introduction of Baluev (2008).

7.6. Algorithmic Considerations

Given the size of the frequency grid required to fully characterize the periods from a given

dataset, it is vital to have an efficient algorithm for evaluating the periodogram. We will see that

– 48 –

the naıve implementation of the standard Lomb-Scargle formula (Equation 33) scales poorly with

the size of the data, but that fast alternatives are available.

Suppose you have a set of N observations over a time-span T for an average cadence of

δt = N/T . From Equation 44 we see that the number of frequencies we need to evaluate is

Nf ∝ Tfmax = Nfmax/δt. Holding constant the average survey cadence δt and fmax, we find

that the number of frequencies required is directly proportional to the number of data points. The

computation of the Lomb-scargle periodogram in Equation 33 requires sums over N sinusoids at

each of the Nf frequencies, and thus we see that the naıve scaling of the algorithm with the size of

the dataset is O(N2), when survey properties are held constant. Due to the trigonometric functions

involved, this turns out to be a rather “expensive” O(N2), which makes the direct implementation

impractical for even modestly-sized datasets.

Fortunately, several faster implementations have been proposed to compute the periodogram

to arbitrary precision in O(N logN) time. The first of these is discussed in Press & Rybicki (1989),

which uses an inverse interpolation operation along with a Fast Fourier Transform to compute

the trigonometric components of Equation 33 very efficiently over a large number of frequencies.

Zechmeister & Kurster (2009) showed how this approach can be easily extended to the floating

mean periodogram discussed in Section 6.2.

A qualitatively similar approach to the Press & Rybicki (1989) algorithm is presented by Leroy

(2012): It makes use of the Non-equispaced Fast Fourier Transform (NFFT, see Keiner et al. 2009)

to compute the Lomb-Scargle periodogram about a factor of 10 faster than the Press & Rybicki

(1989) approach. For the multi-term models discussed in Section 6.3, Palmer (2009) presents

an adaptation of the Press & Rybicki (1989) method that can compute the multi-term result in

O(NK logN), for N data points and K Fourier components.

8. Conclusion and Summary

This paper has been a conceptual tour of the Lomb-Scargle periodogram, from its roots in

Fourier analysis, to its equivalence with special cases of periodic analysis based on least squares

model fitting and Bayesian analysis. From this conceptual understanding, we considered a list of

challenges and issues to be considered when applying the method in practice. We will finish here

with a brief summary of these practical recommendations, along with a somewhat opinionated

post-script for further thought.

8.1. Summary of Recommendations

The previous pages contain a large amount of background and advice for working with the

Lomb-Scargle periodogram. Following is a brief summary of the considerations to keep in mind

– 49 –

when you apply this algorithm to a dataset:

1. Choose an appropriate frequency grid: the minimum can be set to zero, the maximum set

based on the precision of the time measurements (Section 4.1.2), and the grid spacing set

based on the temporal baseline of the data (Section 7.1) so as to not sample too coarsely

around peaks. If this grid size is computationally intractable, reduce the maximum frequency

based on what kinds of signals you are looking for.

2. Compute the window transform using the Lomb-Scargle periodogram, by substituting gn =

1 for each tn and making sure not to pre-center the data or use a floating-mean model

(Section 7.3.1). Examine this window function for dominant features, such as daily or annual

aliases (cf. Figure 13) or Nyquist-like limits (cf. Figure 15).

3. Compute the periodogram for your data. You should always use the floating-mean model

(Section 6.2), as it produces more robust periodograms and has few if any disadvantages.

Avoid multi-term Fourier extensions (Section 6.3) when the signal is unknown, because its

main effect is to increase periodogram noise (cf. Figures 20-21).

4. Plot the periodogram, and identify any patterns that may be caused by features you observed

in the window function power. Plot reference lines showing several False Alarm Probability

levels to understand whether your periodogram peaks are significant enough to be labeled

detections: use the Baluev method or the bootstrap method if it is computationally feasible

(Section 7.4.2). Keep in mind exactly what the False Alarm Probability measures, and avoid

the temptation to misinterpret it (Section 7.4.2).

5. If the window function shows strong aliasing, locate the expected multiple maxima and plot

the phased light curve at each. If there is indication that the sinusoidal model under-fits the

data (cf. Figure 19) then consider re-fitting with a multi-term Fourier model (Section 6.3).

6. If you have prior knowledge of the shape of light curves you are trying to detect, consider using

more complex models to choose between multiple peaks in the periodogram (Section 6.5).

This type of refinement can be quite useful in building automated pipelines for period fitting,

especially in cases where the window aliasing is strong.

7. If you are building an automated pipeline based on Lomb-Scargle for use in a survey, consider

injecting known signals into the pipeline to measure your detection efficiency as a function of

object type, brightness, and other relevant characteristics (Section 7.2)

This list is certainly not comprehensive for all uses of the periodogram, but it should serve as a

brief reminder of the kinds of issues you should keep in mind when using the method to detect

periodic signals.

– 50 –

8.2. Postscript: Why Lomb-Scargle?

After considering all of these practical aspects of the periodogram, I think it is worth stepping

back to revisit the question of why astronomers tend to gravitate toward the Lomb-Scargle approach

rather than the (in many ways simpler) classical periodogram.

As discussed in Section 5, the Lomb-Scargle approach has two distinct benefits over the classical

periodogram: the noise distribution at each individual frequency is chi-square distributed under the

null hypothesis, and the result is equivalent to a periodogram derived from a least squares analysis.

But somehow along the way, a mythology seems to have developed surrounding the efficiency and

efficacy of the Lomb-Scargle approach. In particular, it’s common to see statements or implications

along the lines of the following:

• Myth: The Lomb-Scargle periodogram can be computed more efficiently than the classical

periodogram. Reality: computationally, the two are quite similar, and in fact the fastest

Lomb-Scargle algorithm currently available is based on the classical periodogram computed

via the the NFFT algorithm (see Section 7.6).

• Myth: The Lomb-Scargle periodogram is faster than a direct least squares periodogram because

it avoids explicitly solving for model coefficients. Reality: model coefficients can be determined

with little extra computation (see the discussion in Ivezic et al. 2014).

• Myth: The Lomb-Scargle periodogram allows analytic computation of statistics for periodogram

peaks. Reality: while this is true at individual frequencies, it is not true for the more rel-

evant question of maximum peak heights across multiple frequencies, which must be either

approximated or computed by bootstrap resampling (see Section 7.4)

• Myth: The Lomb-Scargle periodogram corrects for aliasing due to sampling and leads to inde-

pendent noise at each frequency. Reality: for structured window functions common to most

astronomical applications, the Lomb-Scargle periodogram has the same window-driven issues

as the classical periodogram, including spurious peaks due to partial aliasing, and highly

correlated periodogram errors (see Section 7.2).

• Myth: Bayesian analysis shows that Lomb-Scargle is the optimal statistic for detecting periodic

signals in data. Reality: Bayesian analysis shows that Lomb-Scargle is the optimal statistic

for fitting a sinusoid to data, which is not the same as saying it is optimal for finding the

frequency of a generic, potentially non-sinusoidal signal (see Section 6.5).

With these misconceptions corrected, what is the practical advantage of Lomb-Scargle over a classi-

cal periodogram? What would we lose if we instead used the simple classical Fourier periodogram,

estimating uncertainty, significance, and false alarm probability by resampling and simulation, as

we must for Lomb-Scargle itself?

– 51 –

The advantage of analytic statistics for Lomb-Scargle evaporates in light of the need to account

for multiple frequencies, so the only advantage left is the correspondence to least squares and

Bayesian models, and in particular the ability to generalize to more complicated models where

appropriate—but in this case you’re not really using Lomb-Scargle at all, but rather a generative

Bayesian model for your data based on some strong prior information about the form of your

signal. The equivalence of Lomb-Scargle to a Bayesian sinusoidal model is perhaps an interesting

bit of trivia, but not itself a reason to use that model if your data is not known a priori to be

sinusoidal—it could even be construed as an argument against Lomb-Scargle in the general case

where the assumption of a sinusoid is not well-founded.

Conversely, if you replace your Lomb-Scargle approach with a classical periodogram, what you

gain is the ability to reason quantitatively about the effects of the survey window function on the

resulting periodogram (cf. Section 7.3.2). While the deconvolution problem is ill-posed, there is no

reason to assume this is a fatal defect: ill-posed linear models are solved routinely in other areas of

computational research, particularly by using sparsity priors or various forms of regularization. In

any case, I would contend that there is ample room for practitioners to question the prevailing folk

wisdom of the advantage of Lomb-Scargle over approaches based directly on the Fourier transform

and classical periodogram.

8.3. Figures and Code

All computations and figures in this paper were produced using Python, and in particular used

the Numpy (van der Walt et al. 2011; Oliphant 2015), Pandas (McKinney 2010), AstroPy (Astropy

Collaboration et al. 2013), and Matplotlib (Hunter 2007) packages. Periodograms where computed

using the AstroPy implementations9 of the Press & Rybicki (1989), Zechmeister & Kurster (2009),

and Palmer (2009) algorithms, which were adapted from Python code originally published by Ivezic

et al. (2014) and VanderPlas & Ivezic (2015). All code behind this paper, including code to repro-

duce all figures, is available in the form of Jupyter notebooks in the paper’s GitHub repository10.

Acknowledgments: I am grateful to Jeff Scargle, Max Mahlke, Dan Foreman-Mackey, Zeljko

Ivezic, and David Hogg for helpful feedback on early drafts of this paper. This work was supported

by the University of Washington eScience Institute, with funding from the Gordon and Betty Moore

Foundation, the Alfred P. Sloan Foundation, and the Washington Research Foundation.

9The AstroPy Lomb-Scargle documentation is at http://docs.astropy.org/en/stable/stats/lombscargle.

html

10Source code & notebooks available at http://github.com/jakevdp/PracticalLombScargle/

– 52 –

REFERENCES

Anderson, T. 1971, The statistical analysis of time series, Wiley series in probability and mathe-

matical statistics: Probability and mathematical statistics (Wiley)

Angus, R., Foreman-Mackey, D., & Johnson, J. A. 2016, ApJ, 818, 109

Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33

Baluev, R. V. 2008, MNRAS, 385, 1279

Bretthorst, G. 1988, Bayesian Spectrum Analysis and Parameter Estimation, Lecture Notes in

Statistics (Springer)

Bretthorst, G. L. 2001, in American Institute of Physics Conference Series, Vol. 568, Bayesian

Inference and Maximum Entropy Methods in Science and Engineering, ed. A. Mohammad-

Djafari, 241–245

Bretthorst, G. L. 2003, Frequency estimation and generalized Lomb-Scargle periodograms, ed. E. D.

Feigelson & G. J. Babu, 309–329

Cumming, A. 2004, MNRAS, 354, 1165

Cumming, A., Marcy, G. W., & Butler, R. P. 1999, ApJ, 526, 890

Debosscher, J., Sarro, L. M., Aerts, C., et al. 2007, A&A, 475, 1159

Deeming, T. J. 1975, Ap&SS, 36, 137

Delgado, F., Cook, K., Miller, M., Allsman, R., & Pierfederici, F. 2006, in Society of Photo-Optical

Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, Society of Photo-Optical

Instrumentation Engineers (SPIE) Conference Series, 1

Drake, A. J., Graham, M. J., Djorgovski, S. G., et al. 2014, ApJS, 213, 9

Dworetsky, M. M. 1983, MNRAS, 203, 917

Edelson, R. A., & Krolik, J. H. 1988, ApJ, 333, 646

Eyer, L., & Bartholdi, P. 1999, A&AS, 135, 1

Ferraz-Mello, S. 1981, AJ, 86, 619

Foster, G. 1996, AJ, 112, 1709

Frescura, F. A. M., Engelbrecht, C. A., & Frank, B. S. 2008, MNRAS, 388, 1693

Gilliland, R. L., & Baliunas, S. L. 1987, ApJ, 314, 766

– 53 –

Gottlieb, E. W., Wright, E. L., & Liller, W. 1975, ApJ, 195, L33

Graham, M. J., Drake, A. J., Djorgovski, S. G., Mahabal, A. A., & Donalek, C. 2013a, MNRAS,

434, 2629

Graham, M. J., Drake, A. J., Djorgovski, S. G., et al. 2013b, MNRAS, 434, 3423

Gregory, P. C. 2001, in American Institute of Physics Conference Series, Vol. 568, Bayesian In-

ference and Maximum Entropy Methods in Science and Engineering, ed. A. Mohammad-

Djafari, 557–568

Gregory, P. C., & Loredo, T. J. 1992, ApJ, 398, 146

Hilditch, R. W. 2001, An Introduction to Close Binary Stars, 392

Horne, J. H., & Baliunas, S. L. 1986, ApJ, 302, 757

Huijse, P., Estevez, P. A., Protopapas, P., Zegers, P., & Principe, J. C. 2012, IEEE Transactions

on Signal Processing, 60, 5135

Huijse, P., Estevez, P. A., Zegers, P., Principe, J. C., & Protopapas, P. 2011, IEEE Signal Processing

Letters, 18, 371

Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90

Irwin, A. W., Campbell, B., Morbey, C. L., Walker, G. A. H., & Yang, S. 1989, PASP, 101, 147

Ivezic, Z., Connolly, A., VanderPlas, J., & Gray, A. 2014, Statistics, Data Mining, and Machine

Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data, Prince-

ton Series in Modern Observational Astronomy (Princeton University Press)

Ivezic, Z., et al. 2008, arXiv:0805.2366

Jaynes, E. 1987, in Fundamental Theories of Physics, Vol. 21, Maximum-Entropy and Bayesian

Spectral Analysis and Estimation Problems, ed. C. Smith & G. Erickson (Springer Nether-

lands), 1–37

Jaynes, E. T., & Bretthorst, G. L., eds. 2003, Probability theory : the logic of science (Cambridge,

UK, New York: Cambridge University Press)

Keiner, J., Kunis, S., & Potts, D. 2009, ACM Trans. Math. Softw., 36, 19:1

Kelly, B. C., Becker, A. C., Sobolewska, M., Siemiginowska, A., & Uttley, P. 2014, ApJ, 788, 33

Koen, C. 2006, MNRAS, 371, 1390

Kolenberg, K., Szabo, R., Kurtz, D. W., et al. 2010, ApJ, 713, L198

Leroy, B. 2012, A&A, 545, A50

– 54 –

Lomb, N. R. 1976, Ap&SS, 39, 447

Long, J. P., Chi, E. C., & Baraniuk, R. G. 2016, Ann. Appl. Stat., 10, 165

McKinney, W. 2010, in Proceedings of the 9th Python in Science Conference, ed. S. van der Walt

& J. Millman, 51 – 56

McQuillan, A., Aigrain, S., & Roberts, S. 2012, A&A, 539, A137

Mortier, A., Faria, J. P., Correia, C. M., Santerne, A., & Santos, N. C. 2015, A&A, 573, A101

Oliphant, T. E. 2015, Guide to NumPy, 2nd edn. (USA: CreateSpace Independent Publishing

Platform)

Oluseyi, H. M., Becker, A. C., Culliton, C., et al. 2012, AJ, 144, 9

Palaversa, L., Ivezic, Z., Eyer, L., et al. 2013, AJ, 146, 101

Palmer, D. M. 2009, ApJ, 695, 496

Percy, J. R., ed. 1986, The Study of variable stars using small telescopes

Press, W. H., & Rybicki, G. B. 1989, ApJ, 338, 277

Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 2007, Numerical Recipes

3rd Edition: The Art of Scientific Computing, 3rd edn. (New York, NY, USA: Cambridge

University Press)

Reimann, J. D. 1994, PhD thesis, University of California, Berkeley.

Richards, J. W., Starr, D. L., Miller, A. A., et al. 2012, ApJS, 203, 32

Ridgway, S. T., Chandrasekharan, S., Cook, K. H., et al. 2012, in Society of Photo-Optical Instru-

mentation Engineers (SPIE) Conference Series, Vol. 8448, Society of Photo-Optical Instru-

mentation Engineers (SPIE) Conference Series, 10

Roberts, D. H., Lehar, J., & Dreher, J. W. 1987, AJ, 93, 968

Scargle, J. D. 1982, ApJ, 263, 835

—. 1989, ApJ, 343, 874

—. 1998, ApJ, 504, 405

Scargle, J. D. 2002, in American Institute of Physics Conference Series, Vol. 617, Bayesian Inference

and Maximum Entropy Methods in Science and Engineering, ed. R. L. Fry, 23–35

Schuster, A. 1898, Terrestrial Magnetism, 3, 13

– 55 –

Schwarzenberg-Czerny, A. 1989, MNRAS, 241, 153

—. 1996, ApJ, 460, L107

—. 1998, MNRAS, 301, 831

—. 1999, ApJ, 516, 315

Sesar, B., Stuart, J. S., Ivezic, Z., et al. 2011, AJ, 142, 190

Sesar, B., Ivezic, Z., Grammer, S. H., et al. 2010, ApJ, 708, 717

Sesar, B., Grillmair, C. J., Cohen, J. G., et al. 2013, ApJ, 776, 26

Stellingwerf, R. F. 1978, ApJ, 224, 953

Suveges, M. 2012, in Seventh Conference on Astronomical Data Analysis, ed. J.-L. Starck &

C. Surace, 16

Swingler, D. N. 1989, AJ, 97, 280

van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engg., 13, 22

VanderPlas, J. T., & Ivezic, Z. 2015, ArXiv e-prints, arXiv:1502.01344

Vetterli, M., Kovaevi, J., & Goyal, V. K. 2014, Foundations of signal processing (Cambridge:

Cambridge University Press)

Vio, R., Andreani, P., & Biggs, A. 2010, A&A, 519, A85

Wang, Y., Khardon, R., & Protopapas, P. 2012, ApJ, 756, 67

Zechmeister, M., & Kurster, M. 2009, A&A, 496, 577

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